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Theorem dprd2da 20077
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 2735 . 2 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2735 . 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 20040 . . 3 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
64, 5syl 17 . 2 (𝜑𝐺 ∈ Grp)
7 resiun2 6021 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = 𝑖𝐼 (𝐴 ↾ {𝑖})
8 iunid 5065 . . . . . 6 𝑖𝐼 {𝑖} = 𝐼
98reseq2i 5997 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = (𝐴𝐼)
107, 9eqtr3i 2765 . . . 4 𝑖𝐼 (𝐴 ↾ {𝑖}) = (𝐴𝐼)
11 dprd2d.1 . . . . 5 (𝜑 → Rel 𝐴)
12 dprd2d.3 . . . . 5 (𝜑 → dom 𝐴𝐼)
13 relssres 6042 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝐼) → (𝐴𝐼) = 𝐴)
1411, 12, 13syl2anc 584 . . . 4 (𝜑 → (𝐴𝐼) = 𝐴)
1510, 14eqtrid 2787 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) = 𝐴)
16 ovex 7464 . . . . . 6 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
17 eqid 2735 . . . . . 6 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
1816, 17dmmpti 6713 . . . . 5 dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
19 reldmdprd 20032 . . . . . . 7 Rel dom DProd
2019brrelex2i 5746 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V)
21 dmexg 7924 . . . . . 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, 22eqeltrrid 2844 . . . 4 (𝜑𝐼 ∈ V)
24 ressn 6307 . . . . . 6 (𝐴 ↾ {𝑖}) = ({𝑖} × (𝐴 “ {𝑖}))
25 vsnex 5440 . . . . . . 7 {𝑖} ∈ V
26 ovex 7464 . . . . . . . . 9 (𝑖𝑆𝑗) ∈ V
27 eqid 2735 . . . . . . . . 9 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
2826, 27dmmpti 6713 . . . . . . . 8 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
29 dprd2d.4 . . . . . . . . 9 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
3019brrelex2i 5746 . . . . . . . . 9 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V)
31 dmexg 7924 . . . . . . . . 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, 32eqeltrrid 2844 . . . . . . 7 ((𝜑𝑖𝐼) → (𝐴 “ {𝑖}) ∈ V)
34 xpexg 7769 . . . . . . 7 (({𝑖} ∈ V ∧ (𝐴 “ {𝑖}) ∈ V) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3525, 33, 34sylancr 587 . . . . . 6 ((𝜑𝑖𝐼) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3624, 35eqeltrid 2843 . . . . 5 ((𝜑𝑖𝐼) → (𝐴 ↾ {𝑖}) ∈ V)
3736ralrimiva 3144 . . . 4 (𝜑 → ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
38 iunexg 7987 . . . 4 ((𝐼 ∈ V ∧ ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V) → 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
3923, 37, 38syl2anc 584 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
4015, 39eqeltrrd 2840 . 2 (𝜑𝐴 ∈ V)
41 dprd2d.2 . 2 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
42 sneq 4641 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
4342imaeq2d 6080 . . . . . . . . . 10 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
44 oveq1 7438 . . . . . . . . . 10 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
4543, 44mpteq12dv 5239 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
4645breq2d 5160 . . . . . . . 8 (𝑖 = (1st𝑥) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
4729ralrimiva 3144 . . . . . . . . 9 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
4847adantr 480 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
4912adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝐴𝐼)
50 1stdm 8064 . . . . . . . . . 10 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
5111, 50sylan 580 . . . . . . . . 9 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
5249, 51sseldd 3996 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥) ∈ 𝐼)
5346, 48, 52rspcdva 3623 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
54533ad2antr1 1187 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
5554adantr 480 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
56 ovex 7464 . . . . . . 7 ((1st𝑥)𝑆𝑗) ∈ V
57 eqid 2735 . . . . . . 7 (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
5856, 57dmmpti 6713 . . . . . 6 dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)})
5958a1i 11 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)}))
60 1st2nd 8063 . . . . . . . . . . 11 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6111, 60sylan 580 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
62 simpr 484 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
6361, 62eqeltrrd 2840 . . . . . . . . 9 ((𝜑𝑥𝐴) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
64 df-br 5149 . . . . . . . . 9 ((1st𝑥)𝐴(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
6563, 64sylibr 234 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥)𝐴(2nd𝑥))
6611adantr 480 . . . . . . . . 9 ((𝜑𝑥𝐴) → Rel 𝐴)
67 elrelimasn 6106 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
6965, 68mpbird 257 . . . . . . 7 ((𝜑𝑥𝐴) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
70693ad2antr1 1187 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7170adantr 480 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7211adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → Rel 𝐴)
73 simpr2 1194 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦𝐴)
74 1st2nd 8063 . . . . . . . . . . 11 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7572, 73, 74syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7675, 73eqeltrrd 2840 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
77 df-br 5149 . . . . . . . . 9 ((1st𝑦)𝐴(2nd𝑦) ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
7876, 77sylibr 234 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦)𝐴(2nd𝑦))
79 elrelimasn 6106 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8072, 79syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8178, 80mpbird 257 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
8281adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
83 simpr 484 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (1st𝑥) = (1st𝑦))
8483sneqd 4643 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → {(1st𝑥)} = {(1st𝑦)})
8584imaeq2d 6080 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝐴 “ {(1st𝑥)}) = (𝐴 “ {(1st𝑦)}))
8682, 85eleqtrrd 2842 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
87 simplr3 1216 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝑥𝑦)
88 simpr1 1193 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥𝐴)
8972, 88, 60syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
9089, 75eqeq12d 2751 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩))
91 fvex 6920 . . . . . . . . . 10 (1st𝑥) ∈ V
92 fvex 6920 . . . . . . . . . 10 (2nd𝑥) ∈ V
9391, 92opth 5487 . . . . . . . . 9 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9490, 93bitrdi 287 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
9594baibd 539 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥 = 𝑦 ↔ (2nd𝑥) = (2nd𝑦)))
9695necon3bid 2983 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥𝑦 ↔ (2nd𝑥) ≠ (2nd𝑦)))
9787, 96mpbid 232 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ≠ (2nd𝑦))
9855, 59, 71, 86, 97, 1dprdcntz 20043 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ⊆ ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))))
99 df-ov 7434 . . . . . 6 ((1st𝑥)𝑆(2nd𝑥)) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩)
100 oveq2 7439 . . . . . . . 8 (𝑗 = (2nd𝑥) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑥)))
101100, 57, 56fvmpt3i 7021 . . . . . . 7 ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
10270, 101syl 17 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
10389fveq2d 6911 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
10499, 102, 1033eqtr4a 2801 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
105104adantr 480 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
10683oveq1d 7446 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((1st𝑥)𝑆𝑗) = ((1st𝑦)𝑆𝑗))
10785, 106mpteq12dv 5239 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
108107fveq1d 6909 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)))
109 df-ov 7434 . . . . . . . 8 ((1st𝑦)𝑆(2nd𝑦)) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩)
110 oveq2 7439 . . . . . . . . . 10 (𝑗 = (2nd𝑦) → ((1st𝑦)𝑆𝑗) = ((1st𝑦)𝑆(2nd𝑦)))
111 eqid 2735 . . . . . . . . . 10 (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))
112 ovex 7464 . . . . . . . . . 10 ((1st𝑦)𝑆𝑗) ∈ V
113110, 111, 112fvmpt3i 7021 . . . . . . . . 9 ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = ((1st𝑦)𝑆(2nd𝑦)))
11481, 113syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = ((1st𝑦)𝑆(2nd𝑦)))
11575fveq2d 6911 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
116109, 114, 1153eqtr4a 2801 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
117116adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
118108, 117eqtrd 2775 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
119118fveq2d 6911 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))) = ((Cntz‘𝐺)‘(𝑆𝑦)))
12098, 105, 1193sstr3d 4042 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
12111, 41, 12, 29, 4, 3dprd2dlem2 20075 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
12245oveq2d 7447 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
123122, 17, 16fvmpt3i 7021 . . . . . . . 8 ((1st𝑥) ∈ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
12452, 123syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
125121, 124sseqtrrd 4037 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1261253ad2antr1 1187 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
127126adantr 480 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1284ad2antrr 726 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → 𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
12918a1i 11 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
130523ad2antr1 1187 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑥) ∈ 𝐼)
131130adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ∈ 𝐼)
13212adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → dom 𝐴𝐼)
133 1stdm 8064 . . . . . . . . 9 ((Rel 𝐴𝑦𝐴) → (1st𝑦) ∈ dom 𝐴)
13472, 73, 133syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ dom 𝐴)
135132, 134sseldd 3996 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ 𝐼)
136135adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑦) ∈ 𝐼)
137 simpr 484 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ≠ (1st𝑦))
138128, 129, 131, 136, 137, 1dprdcntz 20043 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ⊆ ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))))
139 sneq 4641 . . . . . . . . . . . . 13 (𝑖 = (1st𝑦) → {𝑖} = {(1st𝑦)})
140139imaeq2d 6080 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑦)}))
141 oveq1 7438 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝑖𝑆𝑗) = ((1st𝑦)𝑆𝑗))
142140, 141mpteq12dv 5239 . . . . . . . . . . 11 (𝑖 = (1st𝑦) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
143142oveq2d 7447 . . . . . . . . . 10 (𝑖 = (1st𝑦) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
144143, 17, 16fvmpt3i 7021 . . . . . . . . 9 ((1st𝑦) ∈ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
145135, 144syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦)) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
146145fveq2d 6911 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) = ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))))
147 eqid 2735 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
148147dprdssv 20051 . . . . . . . 8 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))) ⊆ (Base‘𝐺)
149142breq2d 5160 . . . . . . . . . . 11 (𝑖 = (1st𝑦) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
15047adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
151149, 150, 135rspcdva 3623 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
152112, 111dmmpti 6713 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝐴 “ {(1st𝑦)})
153152a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → dom (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝐴 “ {(1st𝑦)}))
154151, 153, 81dprdub 20060 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
155116, 154eqsstrrd 4035 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
156147, 1cntz2ss 19366 . . . . . . . 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)))
157148, 155, 156sylancr 587 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
158146, 157eqsstrd 4034 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
159158adantr 480 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
160138, 159sstrd 4006 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
161127, 160sstrd 4006 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
162120, 161pm2.61dane 3027 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
1636adantr 480 . . . . . 6 ((𝜑𝑥𝐴) → 𝐺 ∈ Grp)
164147subgacs 19192 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
165 acsmre 17697 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
166163, 164, 1653syl 18 . . . . 5 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
16714adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = 𝐴)
168 undif2 4483 . . . . . . . . . . . . . . . . . 18 ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})) = ({(1st𝑥)} ∪ 𝐼)
16952snssd 4814 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → {(1st𝑥)} ⊆ 𝐼)
170 ssequn1 4196 . . . . . . . . . . . . . . . . . . 19 ({(1st𝑥)} ⊆ 𝐼 ↔ ({(1st𝑥)} ∪ 𝐼) = 𝐼)
171169, 170sylib 218 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∪ 𝐼) = 𝐼)
172168, 171eqtr2id 2788 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → 𝐼 = ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})))
173172reseq2d 6000 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
174167, 173eqtr3d 2777 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐴 = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
175 resundi 6014 . . . . . . . . . . . . . . 15 (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))) = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
176174, 175eqtrdi 2791 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
177176difeq1d 4135 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}))
178 difundir 4297 . . . . . . . . . . . . 13 (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
179177, 178eqtrdi 2791 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})))
180 neirr 2947 . . . . . . . . . . . . . . . . 17 ¬ (1st𝑥) ≠ (1st𝑥)
18161eleq1d 2824 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
182 df-br 5149 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
18392brresi 6009 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ((1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}) ∧ (1st𝑥)𝐴(2nd𝑥)))
184183simplbi 497 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}))
185 eldifsni 4795 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}) → (1st𝑥) ≠ (1st𝑥))
186184, 185syl 17 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ≠ (1st𝑥))
187182, 186sylbir 235 . . . . . . . . . . . . . . . . . 18 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥))
188181, 187biimtrdi 253 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥)))
189180, 188mtoi 199 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
190 disjsn 4716 . . . . . . . . . . . . . . . 16 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
191189, 190sylibr 234 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅)
192 disj3 4460 . . . . . . . . . . . . . . 15 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
193191, 192sylib 218 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
194193eqcomd 2741 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}) = (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
195194uneq2d 4178 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
196179, 195eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
197196imaeq2d 6080 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
198 imaundi 6172 . . . . . . . . . 10 (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
199197, 198eqtrdi 2791 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
200199unieqd 4925 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
201 uniun 4935 . . . . . . . 8 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
202200, 201eqtrdi 2791 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
203 imassrn 6091 . . . . . . . . . . 11 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran 𝑆
20441frnd 6745 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
205204adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ (SubGrp‘𝐺))
206 mresspw 17637 . . . . . . . . . . . . 13 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
207166, 206syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
208205, 207sstrd 4006 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
209203, 208sstrid 4007 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
210 sspwuni 5105 . . . . . . . . . 10 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
211209, 210sylib 218 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
212166, 3, 211mrcssidd 17670 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
213 imassrn 6091 . . . . . . . . . . 11 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ ran 𝑆
214213, 208sstrid 4007 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺))
215 sspwuni 5105 . . . . . . . . . 10 ((𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
216214, 215sylib 218 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
217166, 3, 216mrcssidd 17670 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
218 unss12 4198 . . . . . . . 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)}))))))
219212, 217, 218syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
220202, 219eqsstrd 4034 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
2213mrccl 17656 . . . . . . . 8 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
222166, 211, 221syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺))
2233mrccl 17656 . . . . . . . 8 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺))
224166, 216, 223syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺))
225 eqid 2735 . . . . . . . 8 (LSSum‘𝐺) = (LSSum‘𝐺)
226225lsmunss 19692 . . . . . . 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)}))))))
227222, 224, 226syl2anc 584 . . . . . 6 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
228220, 227sstrd 4006 . . . . 5 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
229 difss 4146 . . . . . . . . . . . . 13 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ (𝐴 ↾ {(1st𝑥)})
230 ressn 6307 . . . . . . . . . . . . 13 (𝐴 ↾ {(1st𝑥)}) = ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
231229, 230sseqtri 4032 . . . . . . . . . . . 12 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
232 imass2 6123 . . . . . . . . . . . 12 (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))))
233231, 232ax-mp 5 . . . . . . . . . . 11 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})))
234 ovex 7464 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝑆𝑖) ∈ V
235 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆𝑖))
23657, 235elrnmpt1s 5973 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝐴 “ {(1st𝑥)}) ∧ ((1st𝑥)𝑆𝑖) ∈ V) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
237234, 236mpan2 691 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝐴 “ {(1st𝑥)}) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
238237rgen 3061 . . . . . . . . . . . . . 14 𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
239238a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
240 oveq1 7438 . . . . . . . . . . . . . . . 16 (𝑦 = (1st𝑥) → (𝑦𝑆𝑖) = ((1st𝑥)𝑆𝑖))
241240eleq1d 2824 . . . . . . . . . . . . . . 15 (𝑦 = (1st𝑥) → ((𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
242241ralbidv 3176 . . . . . . . . . . . . . 14 (𝑦 = (1st𝑥) → (∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
24391, 242ralsn 4686 . . . . . . . . . . . . 13 (∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑖 ∈ (𝐴 “ {(1st𝑥)})((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
244239, 243sylibr 234 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
24541adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑆:𝐴⟶(SubGrp‘𝐺))
246245ffund 6741 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun 𝑆)
247 resss 6022 . . . . . . . . . . . . . . 15 (𝐴 ↾ {(1st𝑥)}) ⊆ 𝐴
248230, 247eqsstrri 4031 . . . . . . . . . . . . . 14 ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ 𝐴
249245fdmd 6747 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐴)
250248, 249sseqtrrid 4049 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ dom 𝑆)
251 funimassov 7610 . . . . . . . . . . . . 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))))
252246, 250, 251syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
253244, 252mpbird 257 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
254233, 253sstrid 4007 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
255254unissd 4922 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
256 df-ov 7434 . . . . . . . . . . . . . 14 ((1st𝑥)𝑆𝑗) = (𝑆‘⟨(1st𝑥), 𝑗⟩)
25741ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → 𝑆:𝐴⟶(SubGrp‘𝐺))
258 elrelimasn 6106 . . . . . . . . . . . . . . . . . 18 (Rel 𝐴 → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
25966, 258syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
260259biimpa 476 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (1st𝑥)𝐴𝑗)
261 df-br 5149 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝐴𝑗 ↔ ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
262260, 261sylib 218 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
263257, 262ffvelcdmd 7105 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (𝑆‘⟨(1st𝑥), 𝑗⟩) ∈ (SubGrp‘𝐺))
264256, 263eqeltrid 2843 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ((1st𝑥)𝑆𝑗) ∈ (SubGrp‘𝐺))
265264fmpttd 7135 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)):(𝐴 “ {(1st𝑥)})⟶(SubGrp‘𝐺))
266265frnd 6745 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (SubGrp‘𝐺))
267266, 207sstrd 4006 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺))
268 sspwuni 5105 . . . . . . . . . 10 (ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺) ↔ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
269267, 268sylib 218 . . . . . . . . 9 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
270166, 3, 255, 269mrcssd 17669 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2713dprdspan 20062 . . . . . . . . 9 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) = (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
27253, 271syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) = (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
273270, 272sseqtrrd 4037 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
27416, 17fnmpti 6712 . . . . . . . . . . . . 13 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) Fn 𝐼
275 fnressn 7178 . . . . . . . . . . . . 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))⟩})
276274, 52, 275sylancr 587 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩})
277124opeq2d 4885 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩ = ⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩)
278277sneqd 4643 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩} = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
279276, 278eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
280279oveq2d 7447 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}))
281 dprdsubg 20059 . . . . . . . . . . . . 13 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺))
28253, 281syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ (SubGrp‘𝐺))
283 dprdsn 20071 . . . . . . . . . . . 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)))))
28452, 282, 283syl2anc 584 . . . . . . . . . . 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)))))
285284simprd 495 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
286280, 285eqtrd 2775 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2874adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
28818a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
289 difss 4146 . . . . . . . . . . 11 (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼
290289a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼)
291 disjdif 4478 . . . . . . . . . . 11 ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅
292291a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅)
293287, 288, 169, 290, 292, 1dprdcntz2 20073 . . . . . . . . 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)})))))
294286, 293eqsstrrd 4035 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
29529adantlr 715 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
29666, 245, 49, 295, 287, 3, 290dprd2dlem1 20076 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
297 resmpt 6057 . . . . . . . . . . . 12 ((𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) = (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
298289, 297ax-mp 5 . . . . . . . . . . 11 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})) = (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
299298oveq2i 7442 . . . . . . . . . 10 (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
300296, 299eqtr4di 2793 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))))
301300fveq2d 6911 . . . . . . . 8 ((𝜑𝑥𝐴) → ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
302294, 301sseqtrrd 4037 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
303273, 302sstrd 4006 . . . . . 6 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
304225, 1lsmsubg 19687 . . . . . 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))
305222, 224, 303, 304syl3anc 1370 . . . . 5 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∈ (SubGrp‘𝐺))
3063mrcsscl 17665 . . . . 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)}))))))
307166, 228, 305, 306syl3anc 1370 . . . 4 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ∖ {𝑥}))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
308 sslin 4251 . . . 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)})))))))
309307, 308syl 17 . . 3 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))))
31041ffvelcdmda 7104 . . . 4 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
311225lsmlub 19697 . . . . . . . . . 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)))))
312222, 310, 282, 311syl3anc 1370 . . . . . . . . 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)))))
313273, 121, 312mpbi2and 712 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
314313, 124sseqtrrd 4037 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
315287, 288, 290dprdres 20063 . . . . . . . . . . . 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)))))))
316315simpld 494 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
3173dprdspan 20062 . . . . . . . . . . 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)}))))
318316, 317syl 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)}))))
319 df-ima 5702 . . . . . . . . . . . 12 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
320319unieqi 4924 . . . . . . . . . . 11 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
321320fveq2i 6910 . . . . . . . . . 10 (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
322318, 321eqtr4di 2793 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
323300, 322eqtrd 2775 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
324 eqimss 4054 . . . . . . . 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)}))))
325323, 324syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ⊆ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
326 ss2in 4253 . . . . . . 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)})))))
327314, 325, 326syl2anc 584 . . . . . 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)})))))
328287, 288, 52, 2, 3dprddisj 20044 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∩ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))) = {(0g𝐺)})
329327, 328sseqtrd 4036 . . . . 5 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ {(0g𝐺)})
330225lsmub2 19691 . . . . . . . . 9 (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) ∧ (𝑆𝑥) ∈ (SubGrp‘𝐺)) → (𝑆𝑥) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
331222, 310, 330syl2anc 584 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
3322subg0cl 19165 . . . . . . . . 9 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
333310, 332syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
334331, 333sseldd 3996 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
3352subg0cl 19165 . . . . . . . 8 ((𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
336224, 335syl 17 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
337334, 336elind 4210 . . . . . 6 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
338337snssd 4814 . . . . 5 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
339329, 338eqssd 4013 . . . 4 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = {(0g𝐺)})
340 incom 4217 . . . . 5 ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∩ (𝑆𝑥)) = ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
34169, 101syl 17 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = ((1st𝑥)𝑆(2nd𝑥)))
34261fveq2d 6911 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
34399, 341, 3423eqtr4a 2801 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
344 eqimss2 4055 . . . . . . . . 9 (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥) → (𝑆𝑥) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)))
345343, 344syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)))
346 eldifsn 4791 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥))
34711ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → Rel 𝐴)
348 simprl 771 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐴 ↾ {(1st𝑥)}))
349247, 348sselid 3993 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝐴)
350347, 349, 74syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
351350fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
352351, 109eqtr4di 2793 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑦)𝑆(2nd𝑦)))
353350, 348eqeltrrd 2840 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}))
354 fvex 6920 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑦) ∈ V
355354opelresi 6008 . . . . . . . . . . . . . . . . . . . . 21 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) ↔ ((1st𝑦) ∈ {(1st𝑥)} ∧ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴))
356355simplbi 497 . . . . . . . . . . . . . . . . . . . 20 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) → (1st𝑦) ∈ {(1st𝑥)})
357353, 356syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) ∈ {(1st𝑥)})
358 elsni 4648 . . . . . . . . . . . . . . . . . . 19 ((1st𝑦) ∈ {(1st𝑥)} → (1st𝑦) = (1st𝑥))
359357, 358syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) = (1st𝑥))
360359oveq1d 7446 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ((1st𝑦)𝑆(2nd𝑦)) = ((1st𝑥)𝑆(2nd𝑦)))
361352, 360eqtrd 2775 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑥)𝑆(2nd𝑦)))
362348, 230eleqtrdi 2849 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})))
363 xp2nd 8046 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
364362, 363syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
365 simprr 773 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝑥)
36661adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
367350, 366eqeq12d 2751 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩))
368 fvex 6920 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st𝑦) ∈ V
369368, 354opth 5487 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ ((1st𝑦) = (1st𝑥) ∧ (2nd𝑦) = (2nd𝑥)))
370369baib 535 . . . . . . . . . . . . . . . . . . . . . 22 ((1st𝑦) = (1st𝑥) → (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ (2nd𝑦) = (2nd𝑥)))
371359, 370syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ (2nd𝑦) = (2nd𝑥)))
372367, 371bitrd 279 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ (2nd𝑦) = (2nd𝑥)))
373372necon3bid 2983 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦𝑥 ↔ (2nd𝑦) ≠ (2nd𝑥)))
374365, 373mpbid 232 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ≠ (2nd𝑥))
375 eldifsn 4791 . . . . . . . . . . . . . . . . . 18 ((2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↔ ((2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}) ∧ (2nd𝑦) ≠ (2nd𝑥)))
376364, 374, 375sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))
377 ovex 7464 . . . . . . . . . . . . . . . . 17 ((1st𝑥)𝑆(2nd𝑦)) ∈ V
378 difss 4146 . . . . . . . . . . . . . . . . . . 19 ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ⊆ (𝐴 “ {(1st𝑥)})
379 resmpt 6057 . . . . . . . . . . . . . . . . . . 19 (((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ⊆ (𝐴 “ {(1st𝑥)}) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = (𝑗 ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
380378, 379ax-mp 5 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = (𝑗 ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
381 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (𝑗 = (2nd𝑦) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑦)))
382380, 381elrnmpt1s 5973 . . . . . . . . . . . . . . . . 17 (((2nd𝑦) ∈ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ∧ ((1st𝑥)𝑆(2nd𝑦)) ∈ V) → ((1st𝑥)𝑆(2nd𝑦)) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
383376, 377, 382sylancl 586 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ((1st𝑥)𝑆(2nd𝑦)) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
384361, 383eqeltrd 2839 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
385 df-ima 5702 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))
386384, 385eleqtrrdi 2850 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
387386ex 412 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
388346, 387biimtrid 242 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
389388ralrimiv 3143 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ∀𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})(𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
390231, 250sstrid 4007 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ dom 𝑆)
391 funimass4 6973 . . . . . . . . . . . 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)}))))
392246, 390, 391syl2anc 584 . . . . . . . . . . 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)}))))
393389, 392mpbird 257 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
394393unissd 4922 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
395 imassrn 6091 . . . . . . . . . . 11 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
396395, 267sstrid 4007 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺))
397 sspwuni 5105 . . . . . . . . . 10 (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺) ↔ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
398396, 397sylib 218 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
399166, 3, 394, 398mrcssd 17669 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
400 ss2in 4253 . . . . . . . 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)})))))
401345, 399, 400syl2anc 584 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))))
40258a1i 11 . . . . . . . 8 ((𝜑𝑥𝐴) → dom (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝐴 “ {(1st𝑥)}))
40353, 402, 69, 2, 3dprddisj 20044 . . . . . . 7 ((𝜑𝑥𝐴) → (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))) = {(0g𝐺)})
404401, 403sseqtrd 4036 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ {(0g𝐺)})
4052subg0cl 19165 . . . . . . . . 9 ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
406222, 405syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
407333, 406elind 4210 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
408407snssd 4814 . . . . . 6 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
409404, 408eqssd 4013 . . . . 5 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) = {(0g𝐺)})
410340, 409eqtrid 2787 . . . 4 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∩ (𝑆𝑥)) = {(0g𝐺)})
411225, 222, 310, 224, 2, 339, 410lsmdisj2 19715 . . 3 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))) = {(0g𝐺)})
412309, 411sseqtrd 4036 . 2 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
4131, 2, 3, 6, 40, 41, 162, 412dmdprdd 20034 1 (𝜑𝐺dom DProd 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  cop 4637   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  0gc0g 17486  Moorecmre 17627  mrClscmrc 17628  ACScacs 17630  Grpcgrp 18964  SubGrpcsubg 19151  Cntzccntz 19346  LSSumclsm 19667   DProd cdprd 20028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-gsum 17489  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-lsm 19669  df-cmn 19815  df-dprd 20030
This theorem is referenced by:  dprd2db  20078  dprd2d2  20079
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