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Theorem dprd2da 18373
Description: The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dprd2d.1 (𝜑 → Rel 𝐴)
dprd2d.2 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
dprd2d.3 (𝜑 → dom 𝐴𝐼)
dprd2d.4 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
dprd2d.5 (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
dprd2d.k 𝐾 = (mrCls‘(SubGrp‘𝐺))
Assertion
Ref Expression
dprd2da (𝜑𝐺dom DProd 𝑆)
Distinct variable groups:   𝑖,𝑗,𝐴   𝑖,𝐺,𝑗   𝑖,𝐼   𝑖,𝐾   𝜑,𝑖,𝑗   𝑆,𝑖,𝑗
Allowed substitution hints:   𝐼(𝑗)   𝐾(𝑗)

Proof of Theorem dprd2da
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . 2 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2621 . 2 (0g𝐺) = (0g𝐺)
3 dprd2d.k . 2 𝐾 = (mrCls‘(SubGrp‘𝐺))
4 dprd2d.5 . . 3 (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
5 dprdgrp 18336 . . 3 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
64, 5syl 17 . 2 (𝜑𝐺 ∈ Grp)
7 resiun2 5382 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = 𝑖𝐼 (𝐴 ↾ {𝑖})
8 iunid 4546 . . . . . 6 𝑖𝐼 {𝑖} = 𝐼
98reseq2i 5358 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = (𝐴𝐼)
107, 9eqtr3i 2645 . . . 4 𝑖𝐼 (𝐴 ↾ {𝑖}) = (𝐴𝐼)
11 dprd2d.1 . . . . 5 (𝜑 → Rel 𝐴)
12 dprd2d.3 . . . . 5 (𝜑 → dom 𝐴𝐼)
13 relssres 5401 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝐼) → (𝐴𝐼) = 𝐴)
1411, 12, 13syl2anc 692 . . . 4 (𝜑 → (𝐴𝐼) = 𝐴)
1510, 14syl5eq 2667 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) = 𝐴)
16 ovex 6638 . . . . . 6 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
17 eqid 2621 . . . . . 6 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
1816, 17dmmpti 5985 . . . . 5 dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
19 reldmdprd 18328 . . . . . . 7 Rel dom DProd
2019brrelex2i 5124 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V)
21 dmexg 7051 . . . . . 6 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V)
224, 20, 213syl 18 . . . . 5 (𝜑 → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V)
2318, 22syl5eqelr 2703 . . . 4 (𝜑𝐼 ∈ V)
24 ressn 5635 . . . . . 6 (𝐴 ↾ {𝑖}) = ({𝑖} × (𝐴 “ {𝑖}))
25 snex 4874 . . . . . . 7 {𝑖} ∈ V
26 ovex 6638 . . . . . . . . 9 (𝑖𝑆𝑗) ∈ V
27 eqid 2621 . . . . . . . . 9 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
2826, 27dmmpti 5985 . . . . . . . 8 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
29 dprd2d.4 . . . . . . . . 9 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
3019brrelex2i 5124 . . . . . . . . 9 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V)
31 dmexg 7051 . . . . . . . . 9 ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V)
3229, 30, 313syl 18 . . . . . . . 8 ((𝜑𝑖𝐼) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V)
3328, 32syl5eqelr 2703 . . . . . . 7 ((𝜑𝑖𝐼) → (𝐴 “ {𝑖}) ∈ V)
34 xpexg 6920 . . . . . . 7 (({𝑖} ∈ V ∧ (𝐴 “ {𝑖}) ∈ V) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3525, 33, 34sylancr 694 . . . . . 6 ((𝜑𝑖𝐼) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3624, 35syl5eqel 2702 . . . . 5 ((𝜑𝑖𝐼) → (𝐴 ↾ {𝑖}) ∈ V)
3736ralrimiva 2961 . . . 4 (𝜑 → ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
38 iunexg 7096 . . . 4 ((𝐼 ∈ V ∧ ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V) → 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
3923, 37, 38syl2anc 692 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
4015, 39eqeltrrd 2699 . 2 (𝜑𝐴 ∈ V)
41 dprd2d.2 . 2 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
4212adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝐴𝐼)
43 1stdm 7167 . . . . . . . . . 10 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
4411, 43sylan 488 . . . . . . . . 9 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
4542, 44sseldd 3588 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥) ∈ 𝐼)
4629ralrimiva 2961 . . . . . . . . 9 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
4746adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
48 sneq 4163 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
4948imaeq2d 5430 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
50 oveq1 6617 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
5149, 50mpteq12dv 4698 . . . . . . . . . 10 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
5251breq2d 4630 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
5352rspcv 3294 . . . . . . . 8 ((1st𝑥) ∈ 𝐼 → (∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
5445, 47, 53sylc 65 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
55543ad2antr1 1224 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
5655adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
57 ovex 6638 . . . . . . 7 ((1st𝑥)𝑆𝑗) ∈ V
58 eqid 2621 . . . . . . 7 (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
5957, 58dmmpti 5985 . . . . . 6 dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)})
6059a1i 11 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)}))
61 1st2nd 7166 . . . . . . . . . . 11 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6211, 61sylan 488 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
63 simpr 477 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
6462, 63eqeltrrd 2699 . . . . . . . . 9 ((𝜑𝑥𝐴) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
65 df-br 4619 . . . . . . . . 9 ((1st𝑥)𝐴(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
6664, 65sylibr 224 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥)𝐴(2nd𝑥))
6711adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → Rel 𝐴)
68 elrelimasn 5453 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
6967, 68syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
7066, 69mpbird 247 . . . . . . 7 ((𝜑𝑥𝐴) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
71703ad2antr1 1224 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7271adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7311adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → Rel 𝐴)
74 simpr2 1066 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦𝐴)
75 1st2nd 7166 . . . . . . . . . . 11 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7673, 74, 75syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7776, 74eqeltrrd 2699 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
78 df-br 4619 . . . . . . . . 9 ((1st𝑦)𝐴(2nd𝑦) ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
7977, 78sylibr 224 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦)𝐴(2nd𝑦))
80 elrelimasn 5453 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8173, 80syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8279, 81mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
8382adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
84 simpr 477 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (1st𝑥) = (1st𝑦))
8584sneqd 4165 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → {(1st𝑥)} = {(1st𝑦)})
8685imaeq2d 5430 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝐴 “ {(1st𝑥)}) = (𝐴 “ {(1st𝑦)}))
8783, 86eleqtrrd 2701 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
88 simplr3 1103 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝑥𝑦)
89 simpr1 1065 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥𝐴)
9073, 89, 61syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
9190, 76eqeq12d 2636 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩))
92 fvex 6163 . . . . . . . . . 10 (1st𝑥) ∈ V
93 fvex 6163 . . . . . . . . . 10 (2nd𝑥) ∈ V
9492, 93opth 4910 . . . . . . . . 9 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9591, 94syl6bb 276 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
9695baibd 947 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥 = 𝑦 ↔ (2nd𝑥) = (2nd𝑦)))
9796necon3bid 2834 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥𝑦 ↔ (2nd𝑥) ≠ (2nd𝑦)))
9888, 97mpbid 222 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ≠ (2nd𝑦))
9956, 60, 72, 87, 98, 1dprdcntz 18339 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ⊆ ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))))
100 df-ov 6613 . . . . . 6 ((1st𝑥)𝑆(2nd𝑥)) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩)
101 oveq2 6618 . . . . . . . 8 (𝑗 = (2nd𝑥) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑥)))
102101, 58, 57fvmpt3i 6249 . . . . . . 7 ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
10371, 102syl 17 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
10490fveq2d 6157 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
105100, 103, 1043eqtr4a 2681 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
106105adantr 481 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
10784oveq1d 6625 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((1st𝑥)𝑆𝑗) = ((1st𝑦)𝑆𝑗))
10886, 107mpteq12dv 4698 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
109108fveq1d 6155 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)))
110 df-ov 6613 . . . . . . . 8 ((1st𝑦)𝑆(2nd𝑦)) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩)
111 oveq2 6618 . . . . . . . . . 10 (𝑗 = (2nd𝑦) → ((1st𝑦)𝑆𝑗) = ((1st𝑦)𝑆(2nd𝑦)))
112 eqid 2621 . . . . . . . . . 10 (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))
113 ovex 6638 . . . . . . . . . 10 ((1st𝑦)𝑆𝑗) ∈ V
114111, 112, 113fvmpt3i 6249 . . . . . . . . 9 ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = ((1st𝑦)𝑆(2nd𝑦)))
11582, 114syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = ((1st𝑦)𝑆(2nd𝑦)))
11676fveq2d 6157 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
117110, 115, 1163eqtr4a 2681 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
118117adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
119109, 118eqtrd 2655 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
120119fveq2d 6157 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))) = ((Cntz‘𝐺)‘(𝑆𝑦)))
12199, 106, 1203sstr3d 3631 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
12211, 41, 12, 29, 4, 3dprd2dlem2 18371 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
12351oveq2d 6626 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
124123, 17, 16fvmpt3i 6249 . . . . . . . 8 ((1st𝑥) ∈ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
12545, 124syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
126122, 125sseqtr4d 3626 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1271263ad2antr1 1224 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
128127adantr 481 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1294ad2antrr 761 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → 𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
13018a1i 11 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
131453ad2antr1 1224 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑥) ∈ 𝐼)
132131adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ∈ 𝐼)
13312adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → dom 𝐴𝐼)
134 1stdm 7167 . . . . . . . . 9 ((Rel 𝐴𝑦𝐴) → (1st𝑦) ∈ dom 𝐴)
13573, 74, 134syl2anc 692 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ dom 𝐴)
136133, 135sseldd 3588 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ 𝐼)
137136adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑦) ∈ 𝐼)
138 simpr 477 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ≠ (1st𝑦))
139129, 130, 132, 137, 138, 1dprdcntz 18339 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ⊆ ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))))
140 sneq 4163 . . . . . . . . . . . . 13 (𝑖 = (1st𝑦) → {𝑖} = {(1st𝑦)})
141140imaeq2d 5430 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑦)}))
142 oveq1 6617 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝑖𝑆𝑗) = ((1st𝑦)𝑆𝑗))
143141, 142mpteq12dv 4698 . . . . . . . . . . 11 (𝑖 = (1st𝑦) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
144143oveq2d 6626 . . . . . . . . . 10 (𝑖 = (1st𝑦) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
145144, 17, 16fvmpt3i 6249 . . . . . . . . 9 ((1st𝑦) ∈ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
146136, 145syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
147146fveq2d 6157 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) = ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))))
148 eqid 2621 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
149148dprdssv 18347 . . . . . . . 8 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))) ⊆ (Base‘𝐺)
15046adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
151143breq2d 4630 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
152151rspcv 3294 . . . . . . . . . . 11 ((1st𝑦) ∈ 𝐼 → (∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
153136, 150, 152sylc 65 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
154113, 112dmmpti 5985 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝐴 “ {(1st𝑦)})
155154a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → dom (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝐴 “ {(1st𝑦)}))
156153, 155, 82dprdub 18356 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
157117, 156eqsstr3d 3624 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
158148, 1cntz2ss 17697 . . . . . . . 8 (((𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))) ⊆ (Base‘𝐺) ∧ (𝑆𝑦) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))) → ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
159149, 157, 158sylancr 694 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
160147, 159eqsstrd 3623 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
161160adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
162139, 161sstrd 3597 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
163128, 162sstrd 3597 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
164121, 163pm2.61dane 2877 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
1656adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → 𝐺 ∈ Grp)
166148subgacs 17561 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
167 acsmre 16245 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
168165, 166, 1673syl 18 . . . . 5 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
16914adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = 𝐴)
170 undif2 4021 . . . . . . . . . . . . . . . . . 18 ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})) = ({(1st𝑥)} ∪ 𝐼)
17145snssd 4314 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → {(1st𝑥)} ⊆ 𝐼)
172 ssequn1 3766 . . . . . . . . . . . . . . . . . . 19 ({(1st𝑥)} ⊆ 𝐼 ↔ ({(1st𝑥)} ∪ 𝐼) = 𝐼)
173171, 172sylib 208 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∪ 𝐼) = 𝐼)
174170, 173syl5req 2668 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → 𝐼 = ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})))
175174reseq2d 5361 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
176169, 175eqtr3d 2657 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐴 = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
177 resundi 5374 . . . . . . . . . . . . . . 15 (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))) = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
178176, 177syl6eq 2671 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
179178difeq1d 3710 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}))
180 difundir 3861 . . . . . . . . . . . . 13 (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
181179, 180syl6eq 2671 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})))
182 neirr 2799 . . . . . . . . . . . . . . . . 17 ¬ (1st𝑥) ≠ (1st𝑥)
18362eleq1d 2683 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
184 df-br 4619 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
18593brres 5367 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ((1st𝑥)𝐴(2nd𝑥) ∧ (1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)})))
186185simprbi 480 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}))
187 eldifsni 4294 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}) → (1st𝑥) ≠ (1st𝑥))
188186, 187syl 17 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ≠ (1st𝑥))
189184, 188sylbir 225 . . . . . . . . . . . . . . . . . 18 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥))
190183, 189syl6bi 243 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥)))
191182, 190mtoi 190 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
192 disjsn 4221 . . . . . . . . . . . . . . . 16 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
193191, 192sylibr 224 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅)
194 disj3 3998 . . . . . . . . . . . . . . 15 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
195193, 194sylib 208 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
196195eqcomd 2627 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}) = (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
197196uneq2d 3750 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
198181, 197eqtrd 2655 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
199198imaeq2d 5430 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
200 imaundi 5509 . . . . . . . . . 10 (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
201199, 200syl6eq 2671 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
202201unieqd 4417 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
203 uniun 4427 . . . . . . . 8 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
204202, 203syl6eq 2671 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
205 imassrn 5441 . . . . . . . . . . 11 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran 𝑆
206 frn 6015 . . . . . . . . . . . . . 14 (𝑆:𝐴⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
20741, 206syl 17 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
208207adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ (SubGrp‘𝐺))
209 mresspw 16184 . . . . . . . . . . . . 13 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
210168, 209syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
211208, 210sstrd 3597 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
212205, 211syl5ss 3598 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
213 sspwuni 4582 . . . . . . . . . 10 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
214212, 213sylib 208 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
215168, 3, 214mrcssidd 16217 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
216 imassrn 5441 . . . . . . . . . . 11 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ ran 𝑆
217216, 211syl5ss 3598 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺))
218 sspwuni 4582 . . . . . . . . . 10 ((𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
219217, 218sylib 208 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
220168, 3, 219mrcssidd 16217 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
221 unss12 3768 . . . . . . . 8 (( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∧ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) → ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
222215, 220, 221syl2anc 692 . . . . . . 7 ((𝜑𝑥𝐴) → ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
223204, 222eqsstrd 3623 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
2243mrccl 16203 . . . . . . . 8 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
225168, 214, 224syl2anc 692 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
2263mrccl 16203 . . . . . . . 8 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺))
227168, 219, 226syl2anc 692 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺))
228 eqid 2621 . . . . . . . 8 (LSSum‘𝐺) = (LSSum‘𝐺)
229228lsmunss 18005 . . . . . . 7 (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺)) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
230225, 227, 229syl2anc 692 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
231223, 230sstrd 3597 . . . . 5 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
232 difss 3720 . . . . . . . . . . . . 13 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ (𝐴 ↾ {(1st𝑥)})
233 ressn 5635 . . . . . . . . . . . . 13 (𝐴 ↾ {(1st𝑥)}) = ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
234232, 233sseqtri 3621 . . . . . . . . . . . 12 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
235 imass2 5465 . . . . . . . . . . . 12 (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))))
236234, 235ax-mp 5 . . . . . . . . . . 11 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})))
237 ovex 6638 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝑆𝑖) ∈ V
238 oveq2 6618 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆𝑖))
23958, 238elrnmpt1s 5338 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝐴 “ {(1st𝑥)}) ∧ ((1st𝑥)𝑆𝑖) ∈ V) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
240237, 239mpan2 706 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝐴 “ {(1st𝑥)}) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
241240rgen 2917 . . . . . . . . . . . . . 14 𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
242241a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
243 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑦 = (1st𝑥) → (𝑦𝑆𝑖) = ((1st𝑥)𝑆𝑖))
244243eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑦 = (1st𝑥) → ((𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
245244ralbidv 2981 . . . . . . . . . . . . . 14 (𝑦 = (1st𝑥) → (∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
24692, 245ralsn 4198 . . . . . . . . . . . . 13 (∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
247242, 246sylibr 224 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
24841adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑆:𝐴⟶(SubGrp‘𝐺))
249 ffun 6010 . . . . . . . . . . . . . 14 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
250248, 249syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun 𝑆)
251 resss 5386 . . . . . . . . . . . . . . 15 (𝐴 ↾ {(1st𝑥)}) ⊆ 𝐴
252233, 251eqsstr3i 3620 . . . . . . . . . . . . . 14 ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ 𝐴
253 fdm 6013 . . . . . . . . . . . . . . 15 (𝑆:𝐴⟶(SubGrp‘𝐺) → dom 𝑆 = 𝐴)
254248, 253syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐴)
255252, 254syl5sseqr 3638 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ dom 𝑆)
256 funimassov 6771 . . . . . . . . . . . . 13 ((Fun 𝑆 ∧ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ dom 𝑆) → ((𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
257250, 255, 256syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
258247, 257mpbird 247 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
259236, 258syl5ss 3598 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
260259unissd 4433 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
261 df-ov 6613 . . . . . . . . . . . . . 14 ((1st𝑥)𝑆𝑗) = (𝑆‘⟨(1st𝑥), 𝑗⟩)
26241ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → 𝑆:𝐴⟶(SubGrp‘𝐺))
263 elrelimasn 5453 . . . . . . . . . . . . . . . . . 18 (Rel 𝐴 → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
26467, 263syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
265264biimpa 501 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (1st𝑥)𝐴𝑗)
266 df-br 4619 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝐴𝑗 ↔ ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
267265, 266sylib 208 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
268262, 267ffvelrnd 6321 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (𝑆‘⟨(1st𝑥), 𝑗⟩) ∈ (SubGrp‘𝐺))
269261, 268syl5eqel 2702 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ((1st𝑥)𝑆𝑗) ∈ (SubGrp‘𝐺))
270269, 58fmptd 6346 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)):(𝐴 “ {(1st𝑥)})⟶(SubGrp‘𝐺))
271 frn 6015 . . . . . . . . . . . 12 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)):(𝐴 “ {(1st𝑥)})⟶(SubGrp‘𝐺) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (SubGrp‘𝐺))
272270, 271syl 17 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (SubGrp‘𝐺))
273272, 210sstrd 3597 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺))
274 sspwuni 4582 . . . . . . . . . 10 (ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺) ↔ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
275273, 274sylib 208 . . . . . . . . 9 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
276168, 3, 260, 275mrcssd 16216 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2773dprdspan 18358 . . . . . . . . 9 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) = (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
27854, 277syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) = (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
279276, 278sseqtr4d 3626 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
28016, 17fnmpti 5984 . . . . . . . . . . . . 13 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) Fn 𝐼
281 fnressn 6385 . . . . . . . . . . . . 13 (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) Fn 𝐼 ∧ (1st𝑥) ∈ 𝐼) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩})
282280, 45, 281sylancr 694 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩})
283125opeq2d 4382 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩ = ⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩)
284283sneqd 4165 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩} = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
285282, 284eqtrd 2655 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
286285oveq2d 6626 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}))
287 dprdsubg 18355 . . . . . . . . . . . . 13 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺))
28854, 287syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺))
289 dprdsn 18367 . . . . . . . . . . . 12 (((1st𝑥) ∈ 𝐼 ∧ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺)) → (𝐺dom DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩} ∧ (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))))
29045, 288, 289syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐺dom DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩} ∧ (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))))
291290simprd 479 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
292286, 291eqtrd 2655 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2934adantr 481 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
29418a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
295 difss 3720 . . . . . . . . . . 11 (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼
296295a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼)
297 disjdif 4017 . . . . . . . . . . 11 ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅
298297a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅)
299293, 294, 171, 296, 298, 1dprdcntz2 18369 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
300292, 299eqsstr3d 3624 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
30129adantlr 750 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
30267, 248, 42, 301, 293, 3, 296dprd2dlem1 18372 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
303 resmpt 5413 . . . . . . . . . . . 12 ((𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) = (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
304295, 303ax-mp 5 . . . . . . . . . . 11 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) = (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
305304oveq2i 6621 . . . . . . . . . 10 (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
306302, 305syl6eqr 2673 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))))
307306fveq2d 6157 . . . . . . . 8 ((𝜑𝑥𝐴) → ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
308300, 307sseqtr4d 3626 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
309279, 308sstrd 3597 . . . . . 6 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
310228, 1lsmsubg 18001 . . . . . 6 (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺) ∧ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∈ (SubGrp‘𝐺))
311225, 227, 309, 310syl3anc 1323 . . . . 5 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∈ (SubGrp‘𝐺))
3123mrcsscl 16212 . . . . 5 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∧ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∈ (SubGrp‘𝐺)) → (𝐾 (𝑆 “ (𝐴 ∖ {𝑥}))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
313168, 231, 311, 312syl3anc 1323 . . . 4 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ∖ {𝑥}))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
314 sslin 3822 . . . 4 ((𝐾 (𝑆 “ (𝐴 ∖ {𝑥}))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))))
315313, 314syl 17 . . 3 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))))
31641ffvelrnda 6320 . . . 4 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
317228lsmlub 18010 . . . . . . . . . 10 (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ (𝑆𝑥) ∈ (SubGrp‘𝐺) ∧ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺)) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∧ (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))) ↔ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))))
318225, 316, 288, 317syl3anc 1323 . . . . . . . . 9 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∧ (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))) ↔ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))))
319279, 122, 318mpbi2and 955 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
320319, 125sseqtr4d 3626 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
321293, 294, 296dprdres 18359 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) ∧ (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))))
322321simpld 475 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
3233dprdspan 18358 . . . . . . . . . . 11 (𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))))
324322, 323syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))))
325 df-ima 5092 . . . . . . . . . . . 12 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
326325unieqi 4416 . . . . . . . . . . 11 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
327326fveq2i 6156 . . . . . . . . . 10 (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
328324, 327syl6eqr 2673 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
329306, 328eqtrd 2655 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
330 eqimss 3641 . . . . . . . 8 ((𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
331329, 330syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
332 ss2in 3823 . . . . . . 7 ((((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∧ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∩ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))))
333320, 331, 332syl2anc 692 . . . . . 6 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∩ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))))
334293, 294, 45, 2, 3dprddisj 18340 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∩ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))) = {(0g𝐺)})
335333, 334sseqtrd 3625 . . . . 5 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ {(0g𝐺)})
336228lsmub2 18004 . . . . . . . . 9 (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ (𝑆𝑥) ∈ (SubGrp‘𝐺)) → (𝑆𝑥) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
337225, 316, 336syl2anc 692 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
3382subg0cl 17534 . . . . . . . . 9 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
339316, 338syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
340337, 339sseldd 3588 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
3412subg0cl 17534 . . . . . . . 8 ((𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
342227, 341syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
343340, 342elind 3781 . . . . . 6 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
344343snssd 4314 . . . . 5 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
345335, 344eqssd 3604 . . . 4 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = {(0g𝐺)})
346 incom 3788 . . . . 5 ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∩ (𝑆𝑥)) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
34770, 102syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
34862fveq2d 6157 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
349100, 347, 3483eqtr4a 2681 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
350 eqimss2 3642 . . . . . . . . 9 (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥) → (𝑆𝑥) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)))
351349, 350syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)))
352 eldifsn 4292 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥))
35311ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → Rel 𝐴)
354 simprl 793 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐴 ↾ {(1st𝑥)}))
355251, 354sseldi 3585 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝐴)
356353, 355, 75syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
357356fveq2d 6157 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
358357, 110syl6eqr 2673 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑦)𝑆(2nd𝑦)))
359356, 354eqeltrrd 2699 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}))
360 fvex 6163 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑦) ∈ V
361360opelres 5366 . . . . . . . . . . . . . . . . . . . . 21 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) ↔ (⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴 ∧ (1st𝑦) ∈ {(1st𝑥)}))
362361simprbi 480 . . . . . . . . . . . . . . . . . . . 20 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) → (1st𝑦) ∈ {(1st𝑥)})
363359, 362syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) ∈ {(1st𝑥)})
364 elsni 4170 . . . . . . . . . . . . . . . . . . 19 ((1st𝑦) ∈ {(1st𝑥)} → (1st𝑦) = (1st𝑥))
365363, 364syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) = (1st𝑥))
366365oveq1d 6625 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ((1st𝑦)𝑆(2nd𝑦)) = ((1st𝑥)𝑆(2nd𝑦)))
367358, 366eqtrd 2655 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑥)𝑆(2nd𝑦)))
368354, 233syl6eleq 2708 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})))
369 xp2nd 7151 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
370368, 369syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
371 simprr 795 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝑥)
37262adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
373356, 372eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩))
374 fvex 6163 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st𝑦) ∈ V
375374, 360opth 4910 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ ((1st𝑦) = (1st𝑥) ∧ (2nd𝑦) = (2nd𝑥)))
376375baib 943 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑦) = (1st𝑥) → (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ (2nd𝑦) = (2nd𝑥)))
377365, 376syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ (2nd𝑦) = (2nd𝑥)))
378373, 377bitrd 268 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ (2nd𝑦) = (2nd𝑥)))
379378necon3bid 2834 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦𝑥 ↔ (2nd𝑦) ≠ (2nd𝑥)))
380371, 379mpbid 222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ≠ (2nd𝑥))
381 eldifsn 4292 . . . . . . . . . . . . . . . . . 18 ((2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↔ ((2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}) ∧ (2nd𝑦) ≠ (2nd𝑥)))
382370, 380, 381sylanbrc 697 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))
383 ovex 6638 . . . . . . . . . . . . . . . . 17 ((1st𝑥)𝑆(2nd𝑦)) ∈ V
384 difss 3720 . . . . . . . . . . . . . . . . . . 19 ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ⊆ (𝐴 “ {(1st𝑥)})
385 resmpt 5413 . . . . . . . . . . . . . . . . . . 19 (((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ⊆ (𝐴 “ {(1st𝑥)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = (𝑗 ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
386384, 385ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = (𝑗 ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
387 oveq2 6618 . . . . . . . . . . . . . . . . . 18 (𝑗 = (2nd𝑦) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑦)))
388386, 387elrnmpt1s 5338 . . . . . . . . . . . . . . . . 17 (((2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ∧ ((1st𝑥)𝑆(2nd𝑦)) ∈ V) → ((1st𝑥)𝑆(2nd𝑦)) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
389382, 383, 388sylancl 693 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ((1st𝑥)𝑆(2nd𝑦)) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
390367, 389eqeltrd 2698 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
391 df-ima 5092 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))
392390, 391syl6eleqr 2709 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
393392ex 450 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
394352, 393syl5bi 232 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
395394ralrimiv 2960 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ∀𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})(𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
396234, 255syl5ss 3598 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ dom 𝑆)
397 funimass4 6209 . . . . . . . . . . . 12 ((Fun 𝑆 ∧ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ dom 𝑆) → ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ↔ ∀𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})(𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
398250, 396, 397syl2anc 692 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ↔ ∀𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})(𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
399395, 398mpbird 247 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
400399unissd 4433 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
401 imassrn 5441 . . . . . . . . . . 11 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
402401, 273syl5ss 3598 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺))
403 sspwuni 4582 . . . . . . . . . 10 (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺) ↔ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
404402, 403sylib 208 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
405168, 3, 400, 404mrcssd 16216 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
406 ss2in 3823 . . . . . . . 8 (((𝑆𝑥) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∧ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))))
407351, 405, 406syl2anc 692 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))))
40859a1i 11 . . . . . . . 8 ((𝜑𝑥𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)}))
40954, 408, 70, 2, 3dprddisj 18340 . . . . . . 7 ((𝜑𝑥𝐴) → (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))) = {(0g𝐺)})
410407, 409sseqtrd 3625 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ {(0g𝐺)})
4112subg0cl 17534 . . . . . . . . 9 ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
412225, 411syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
413339, 412elind 3781 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
414413snssd 4314 . . . . . 6 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
415410, 414eqssd 3604 . . . . 5 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) = {(0g𝐺)})
416346, 415syl5eq 2667 . . . 4 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∩ (𝑆𝑥)) = {(0g𝐺)})
417228, 225, 316, 227, 2, 345, 416lsmdisj2 18027 . . 3 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))) = {(0g𝐺)})
418315, 417sseqtrd 3625 . 2 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
4191, 2, 3, 6, 40, 41, 164, 418dmdprdd 18330 1 (𝜑𝐺dom DProd 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3189  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135  {csn 4153  cop 4159   cuni 4407   ciun 4490   class class class wbr 4618  cmpt 4678   × cxp 5077  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Rel wrel 5084  Fun wfun 5846   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  Basecbs 15792  0gc0g 16032  Moorecmre 16174  mrClscmrc 16175  ACScacs 16177  Grpcgrp 17354  SubGrpcsubg 17520  Cntzccntz 17680  LSSumclsm 17981   DProd cdprd 18324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-tpos 7304  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-oi 8367  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-gsum 16035  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-grp 17357  df-minusg 17358  df-sbg 17359  df-mulg 17473  df-subg 17523  df-ghm 17590  df-gim 17633  df-cntz 17682  df-oppg 17708  df-lsm 17983  df-cmn 18127  df-dprd 18326
This theorem is referenced by:  dprd2db  18374  dprd2d2  18375
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