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Theorem dprd2da 18886
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 2795 . 2 (Cntz‘𝐺) = (Cntz‘𝐺)
2 eqid 2795 . 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 18849 . . 3 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
64, 5syl 17 . 2 (𝜑𝐺 ∈ Grp)
7 resiun2 5760 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = 𝑖𝐼 (𝐴 ↾ {𝑖})
8 iunid 4887 . . . . . 6 𝑖𝐼 {𝑖} = 𝐼
98reseq2i 5736 . . . . 5 (𝐴 𝑖𝐼 {𝑖}) = (𝐴𝐼)
107, 9eqtr3i 2821 . . . 4 𝑖𝐼 (𝐴 ↾ {𝑖}) = (𝐴𝐼)
11 dprd2d.1 . . . . 5 (𝜑 → Rel 𝐴)
12 dprd2d.3 . . . . 5 (𝜑 → dom 𝐴𝐼)
13 relssres 5779 . . . . 5 ((Rel 𝐴 ∧ dom 𝐴𝐼) → (𝐴𝐼) = 𝐴)
1411, 12, 13syl2anc 584 . . . 4 (𝜑 → (𝐴𝐼) = 𝐴)
1510, 14syl5eq 2843 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) = 𝐴)
16 ovex 7053 . . . . . 6 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
17 eqid 2795 . . . . . 6 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
1816, 17dmmpti 6365 . . . . 5 dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
19 reldmdprd 18841 . . . . . . 7 Rel dom DProd
2019brrelex2i 5500 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ∈ V)
21 dmexg 7474 . . . . . 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 2888 . . . 4 (𝜑𝐼 ∈ V)
24 ressn 6016 . . . . . 6 (𝐴 ↾ {𝑖}) = ({𝑖} × (𝐴 “ {𝑖}))
25 snex 5228 . . . . . . 7 {𝑖} ∈ V
26 ovex 7053 . . . . . . . . 9 (𝑖𝑆𝑗) ∈ V
27 eqid 2795 . . . . . . . . 9 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
2826, 27dmmpti 6365 . . . . . . . 8 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
29 dprd2d.4 . . . . . . . . 9 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
3019brrelex2i 5500 . . . . . . . . 9 (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ∈ V)
31 dmexg 7474 . . . . . . . . 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 2888 . . . . . . 7 ((𝜑𝑖𝐼) → (𝐴 “ {𝑖}) ∈ V)
34 xpexg 7335 . . . . . . 7 (({𝑖} ∈ V ∧ (𝐴 “ {𝑖}) ∈ V) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3525, 33, 34sylancr 587 . . . . . 6 ((𝜑𝑖𝐼) → ({𝑖} × (𝐴 “ {𝑖})) ∈ V)
3624, 35syl5eqel 2887 . . . . 5 ((𝜑𝑖𝐼) → (𝐴 ↾ {𝑖}) ∈ V)
3736ralrimiva 3149 . . . 4 (𝜑 → ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
38 iunexg 7525 . . . 4 ((𝐼 ∈ V ∧ ∀𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V) → 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
3923, 37, 38syl2anc 584 . . 3 (𝜑 𝑖𝐼 (𝐴 ↾ {𝑖}) ∈ V)
4015, 39eqeltrrd 2884 . 2 (𝜑𝐴 ∈ V)
41 dprd2d.2 . 2 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
42 sneq 4486 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
4342imaeq2d 5811 . . . . . . . . . 10 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
44 oveq1 7028 . . . . . . . . . 10 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
4543, 44mpteq12dv 5050 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
4645breq2d 4978 . . . . . . . 8 (𝑖 = (1st𝑥) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
4729ralrimiva 3149 . . . . . . . . 9 (𝜑 → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
4847adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
4912adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → dom 𝐴𝐼)
50 1stdm 7600 . . . . . . . . . 10 ((Rel 𝐴𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
5111, 50sylan 580 . . . . . . . . 9 ((𝜑𝑥𝐴) → (1st𝑥) ∈ dom 𝐴)
5249, 51sseldd 3894 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥) ∈ 𝐼)
5346, 48, 52rspcdva 3565 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
54533ad2antr1 1181 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
5554adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
56 ovex 7053 . . . . . . 7 ((1st𝑥)𝑆𝑗) ∈ V
57 eqid 2795 . . . . . . 7 (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
5856, 57dmmpti 6365 . . . . . 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 7599 . . . . . . . . . . 11 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
6111, 60sylan 580 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
62 simpr 485 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥𝐴)
6361, 62eqeltrrd 2884 . . . . . . . . 9 ((𝜑𝑥𝐴) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
64 df-br 4967 . . . . . . . . 9 ((1st𝑥)𝐴(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴)
6563, 64sylibr 235 . . . . . . . 8 ((𝜑𝑥𝐴) → (1st𝑥)𝐴(2nd𝑥))
6611adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐴) → Rel 𝐴)
67 elrelimasn 5834 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
6866, 67syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → ((2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴(2nd𝑥)))
6965, 68mpbird 258 . . . . . . 7 ((𝜑𝑥𝐴) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
70693ad2antr1 1181 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7170adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ∈ (𝐴 “ {(1st𝑥)}))
7211adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → Rel 𝐴)
73 simpr2 1188 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦𝐴)
74 1st2nd 7599 . . . . . . . . . . 11 ((Rel 𝐴𝑦𝐴) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7572, 73, 74syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
7675, 73eqeltrrd 2884 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
77 df-br 4967 . . . . . . . . 9 ((1st𝑦)𝐴(2nd𝑦) ↔ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴)
7876, 77sylibr 235 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦)𝐴(2nd𝑦))
79 elrelimasn 5834 . . . . . . . . 9 (Rel 𝐴 → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8072, 79syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}) ↔ (1st𝑦)𝐴(2nd𝑦)))
8178, 80mpbird 258 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
8281adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑦)}))
83 simpr 485 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (1st𝑥) = (1st𝑦))
8483sneqd 4488 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → {(1st𝑥)} = {(1st𝑦)})
8584imaeq2d 5811 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝐴 “ {(1st𝑥)}) = (𝐴 “ {(1st𝑦)}))
8682, 85eleqtrrd 2886 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
87 simplr3 1210 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → 𝑥𝑦)
88 simpr1 1187 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥𝐴)
8972, 88, 60syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
9089, 75eqeq12d 2810 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩))
91 fvex 6556 . . . . . . . . . 10 (1st𝑥) ∈ V
92 fvex 6556 . . . . . . . . . 10 (2nd𝑥) ∈ V
9391, 92opth 5265 . . . . . . . . 9 (⟨(1st𝑥), (2nd𝑥)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩ ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦)))
9490, 93syl6bb 288 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑥 = 𝑦 ↔ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) = (2nd𝑦))))
9594baibd 540 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥 = 𝑦 ↔ (2nd𝑥) = (2nd𝑦)))
9695necon3bid 3028 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑥𝑦 ↔ (2nd𝑥) ≠ (2nd𝑦)))
9787, 96mpbid 233 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (2nd𝑥) ≠ (2nd𝑦))
9855, 59, 71, 86, 97, 1dprdcntz 18852 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ⊆ ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))))
99 df-ov 7024 . . . . . 6 ((1st𝑥)𝑆(2nd𝑥)) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩)
100 oveq2 7029 . . . . . . . 8 (𝑗 = (2nd𝑥) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑥)))
101100, 57, 56fvmpt3i 6645 . . . . . . 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 6547 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
10499, 102, 1033eqtr4a 2857 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
105104adantr 481 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
10683oveq1d 7036 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((1st𝑥)𝑆𝑗) = ((1st𝑦)𝑆𝑗))
10785, 106mpteq12dv 5050 . . . . . . 7 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
108107fveq1d 6545 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)))
109 df-ov 7024 . . . . . . . 8 ((1st𝑦)𝑆(2nd𝑦)) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩)
110 oveq2 7029 . . . . . . . . . 10 (𝑗 = (2nd𝑦) → ((1st𝑦)𝑆𝑗) = ((1st𝑦)𝑆(2nd𝑦)))
111 eqid 2795 . . . . . . . . . 10 (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))
112 ovex 7053 . . . . . . . . . 10 ((1st𝑦)𝑆𝑗) ∈ V
113110, 111, 112fvmpt3i 6645 . . . . . . . . 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 6547 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
116109, 114, 1153eqtr4a 2857 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
117116adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
118108, 117eqtrd 2831 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦)) = (𝑆𝑦))
119118fveq2d 6547 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → ((Cntz‘𝐺)‘((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑦))) = ((Cntz‘𝐺)‘(𝑆𝑦)))
12098, 105, 1193sstr3d 3938 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) = (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
12111, 41, 12, 29, 4, 3dprd2dlem2 18884 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
12245oveq2d 7037 . . . . . . . . 9 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
123122, 17, 16fvmpt3i 6645 . . . . . . . 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, 124sseqtr4d 3933 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1261253ad2antr1 1181 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
127126adantr 481 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
1284ad2antrr 722 . . . . . 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 1181 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑥) ∈ 𝐼)
131130adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ∈ 𝐼)
13212adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → dom 𝐴𝐼)
133 1stdm 7600 . . . . . . . . 9 ((Rel 𝐴𝑦𝐴) → (1st𝑦) ∈ dom 𝐴)
13472, 73, 133syl2anc 584 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ dom 𝐴)
135132, 134sseldd 3894 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (1st𝑦) ∈ 𝐼)
136135adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑦) ∈ 𝐼)
137 simpr 485 . . . . . 6 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (1st𝑥) ≠ (1st𝑦))
138128, 129, 131, 136, 137, 1dprdcntz 18852 . . . . 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 4486 . . . . . . . . . . . . 13 (𝑖 = (1st𝑦) → {𝑖} = {(1st𝑦)})
140139imaeq2d 5811 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑦)}))
141 oveq1 7028 . . . . . . . . . . . 12 (𝑖 = (1st𝑦) → (𝑖𝑆𝑗) = ((1st𝑦)𝑆𝑗))
142140, 141mpteq12dv 5050 . . . . . . . . . . 11 (𝑖 = (1st𝑦) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
143142oveq2d 7037 . . . . . . . . . 10 (𝑖 = (1st𝑦) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
144143, 17, 16fvmpt3i 6645 . . . . . . . . 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 6547 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) = ((Cntz‘𝐺)‘(𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))))
147 eqid 2795 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
148147dprdssv 18860 . . . . . . . 8 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))) ⊆ (Base‘𝐺)
149142breq2d 4978 . . . . . . . . . . 11 (𝑖 = (1st𝑦) → (𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) ↔ 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
15047adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ∀𝑖𝐼 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
151149, 150, 135rspcdva 3565 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗)))
152112, 111dmmpti 6365 . . . . . . . . . . 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 18869 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))‘(2nd𝑦)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
155116, 154eqsstrrd 3931 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑦) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑦)}) ↦ ((1st𝑦)𝑆𝑗))))
156147, 1cntz2ss 18209 . . . . . . . 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 3930 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
159158adantr 481 . . . . 5 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((Cntz‘𝐺)‘((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑦))) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
160138, 159sstrd 3903 . . . 4 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
161127, 160sstrd 3903 . . 3 (((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) ∧ (1st𝑥) ≠ (1st𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
162120, 161pm2.61dane 3072 . 2 ((𝜑 ∧ (𝑥𝐴𝑦𝐴𝑥𝑦)) → (𝑆𝑥) ⊆ ((Cntz‘𝐺)‘(𝑆𝑦)))
1636adantr 481 . . . . . 6 ((𝜑𝑥𝐴) → 𝐺 ∈ Grp)
164147subgacs 18073 . . . . . 6 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
165 acsmre 16757 . . . . . 6 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
166163, 164, 1653syl 18 . . . . 5 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
16714adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = 𝐴)
168 undif2 4343 . . . . . . . . . . . . . . . . . 18 ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})) = ({(1st𝑥)} ∪ 𝐼)
16952snssd 4653 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝐴) → {(1st𝑥)} ⊆ 𝐼)
170 ssequn1 4081 . . . . . . . . . . . . . . . . . . 19 ({(1st𝑥)} ⊆ 𝐼 ↔ ({(1st𝑥)} ∪ 𝐼) = 𝐼)
171169, 170sylib 219 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∪ 𝐼) = 𝐼)
172168, 171syl5req 2844 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → 𝐼 = ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)})))
173172reseq2d 5739 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → (𝐴𝐼) = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
174167, 173eqtr3d 2833 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → 𝐴 = (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))))
175 resundi 5753 . . . . . . . . . . . . . . 15 (𝐴 ↾ ({(1st𝑥)} ∪ (𝐼 ∖ {(1st𝑥)}))) = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
176174, 175syl6eq 2847 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝐴 = ((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
177176difeq1d 4023 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}))
178 difundir 4181 . . . . . . . . . . . . 13 (((𝐴 ↾ {(1st𝑥)}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
179177, 178syl6eq 2847 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})))
180 neirr 2993 . . . . . . . . . . . . . . . . 17 ¬ (1st𝑥) ≠ (1st𝑥)
18161eleq1d 2867 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
182 df-br 4967 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
18392brresi 5748 . . . . . . . . . . . . . . . . . . . . 21 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) ↔ ((1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}) ∧ (1st𝑥)𝐴(2nd𝑥)))
184183simplbi 498 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}))
185 eldifsni 4633 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑥) ∈ (𝐼 ∖ {(1st𝑥)}) → (1st𝑥) ≠ (1st𝑥))
186184, 185syl 17 . . . . . . . . . . . . . . . . . . 19 ((1st𝑥)(𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))(2nd𝑥) → (1st𝑥) ≠ (1st𝑥))
187182, 186sylbir 236 . . . . . . . . . . . . . . . . . 18 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥))
188181, 187syl6bi 254 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) → (1st𝑥) ≠ (1st𝑥)))
189180, 188mtoi 200 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝐴) → ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
190 disjsn 4558 . . . . . . . . . . . . . . . 16 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
191189, 190sylibr 235 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅)
192 disj3 4321 . . . . . . . . . . . . . . 15 (((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∩ {𝑥}) = ∅ ↔ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
193191, 192sylib 219 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) = ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}))
194193eqcomd 2801 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥}) = (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))
195194uneq2d 4064 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ ((𝐴 ↾ (𝐼 ∖ {(1st𝑥)})) ∖ {𝑥})) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
196179, 195eqtrd 2831 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐴 ∖ {𝑥}) = (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
197196imaeq2d 5811 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
198 imaundi 5889 . . . . . . . . . 10 (𝑆 “ (((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ∪ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
199197, 198syl6eq 2847 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
200199unieqd 4759 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
201 uniun 4768 . . . . . . . 8 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))
202200, 201syl6eq 2847 . . . . . . 7 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) = ( (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ∪ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
203 imassrn 5822 . . . . . . . . . . 11 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran 𝑆
20441frnd 6394 . . . . . . . . . . . . 13 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
205204adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ (SubGrp‘𝐺))
206 mresspw 16697 . . . . . . . . . . . . 13 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
207166, 206syl 17 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
208205, 207sstrd 3903 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
209203, 208sstrid 3904 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺))
210 sspwuni 4925 . . . . . . . . . 10 ((𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
211209, 210sylib 219 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (Base‘𝐺))
212166, 3, 211mrcssidd 16730 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))))
213 imassrn 5822 . . . . . . . . . . 11 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ ran 𝑆
214213, 208sstrid 3904 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺))
215 sspwuni 4925 . . . . . . . . . 10 ((𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
216214, 215sylib 219 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (Base‘𝐺))
217166, 3, 216mrcssidd 16730 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))) ⊆ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))
218 unss12 4083 . . . . . . . 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 3930 . . . . . 6 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∪ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
2213mrccl 16716 . . . . . . . 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 16716 . . . . . . . 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 2795 . . . . . . . 8 (LSSum‘𝐺) = (LSSum‘𝐺)
226225lsmunss 18518 . . . . . . 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 3903 . . . . 5 ((𝜑𝑥𝐴) → (𝑆 “ (𝐴 ∖ {𝑥})) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
229 difss 4033 . . . . . . . . . . . . 13 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ (𝐴 ↾ {(1st𝑥)})
230 ressn 6016 . . . . . . . . . . . . 13 (𝐴 ↾ {(1st𝑥)}) = ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
231229, 230sseqtri 3928 . . . . . . . . . . . 12 ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))
232 imass2 5846 . . . . . . . . . . . 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 7053 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝑆𝑖) ∈ V
235 oveq2 7029 . . . . . . . . . . . . . . . . 17 (𝑗 = 𝑖 → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆𝑖))
23657, 235elrnmpt1s 5716 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (𝐴 “ {(1st𝑥)}) ∧ ((1st𝑥)𝑆𝑖) ∈ V) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
237234, 236mpan2 687 . . . . . . . . . . . . . . 15 (𝑖 ∈ (𝐴 “ {(1st𝑥)}) → ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
238237rgen 3115 . . . . . . . . . . . . . 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 7028 . . . . . . . . . . . . . . . 16 (𝑦 = (1st𝑥) → (𝑦𝑆𝑖) = ((1st𝑥)𝑆𝑖))
241240eleq1d 2867 . . . . . . . . . . . . . . 15 (𝑦 = (1st𝑥) → ((𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↔ ((1st𝑥)𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
242241ralbidv 3164 . . . . . . . . . . . . . 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 4530 . . . . . . . . . . . . 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 235 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ∀𝑦 ∈ {(1st𝑥)}∀𝑖 ∈ (𝐴 “ {(1st𝑥)})(𝑦𝑆𝑖) ∈ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
24541adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → 𝑆:𝐴⟶(SubGrp‘𝐺))
246245ffund 6391 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → Fun 𝑆)
247 resss 5764 . . . . . . . . . . . . . . 15 (𝐴 ↾ {(1st𝑥)}) ⊆ 𝐴
248230, 247eqsstrri 3927 . . . . . . . . . . . . . 14 ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ 𝐴
249245fdmd 6396 . . . . . . . . . . . . . 14 ((𝜑𝑥𝐴) → dom 𝑆 = 𝐴)
250248, 249sseqtrrid 3945 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) ⊆ dom 𝑆)
251 funimassov 7186 . . . . . . . . . . . . 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 258 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝑆 “ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)}))) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
254233, 253sstrid 3904 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
255254unissd 4773 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
256 df-ov 7024 . . . . . . . . . . . . . 14 ((1st𝑥)𝑆𝑗) = (𝑆‘⟨(1st𝑥), 𝑗⟩)
25741ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → 𝑆:𝐴⟶(SubGrp‘𝐺))
258 elrelimasn 5834 . . . . . . . . . . . . . . . . . 18 (Rel 𝐴 → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
25966, 258syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↔ (1st𝑥)𝐴𝑗))
260259biimpa 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (1st𝑥)𝐴𝑗)
261 df-br 4967 . . . . . . . . . . . . . . . 16 ((1st𝑥)𝐴𝑗 ↔ ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
262260, 261sylib 219 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ⟨(1st𝑥), 𝑗⟩ ∈ 𝐴)
263257, 262ffvelrnd 6722 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → (𝑆‘⟨(1st𝑥), 𝑗⟩) ∈ (SubGrp‘𝐺))
264256, 263syl5eqel 2887 . . . . . . . . . . . . 13 (((𝜑𝑥𝐴) ∧ 𝑗 ∈ (𝐴 “ {(1st𝑥)})) → ((1st𝑥)𝑆𝑗) ∈ (SubGrp‘𝐺))
265264fmpttd 6747 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)):(𝐴 “ {(1st𝑥)})⟶(SubGrp‘𝐺))
266265frnd 6394 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (SubGrp‘𝐺))
267266, 207sstrd 3903 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺))
268 sspwuni 4925 . . . . . . . . . 10 (ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ 𝒫 (Base‘𝐺) ↔ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
269267, 268sylib 219 . . . . . . . . 9 ((𝜑𝑥𝐴) → ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ⊆ (Base‘𝐺))
270166, 3, 255, 269mrcssd 16729 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2713dprdspan 18871 . . . . . . . . 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, 272sseqtr4d 3933 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
27416, 17fnmpti 6364 . . . . . . . . . . . . 13 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) Fn 𝐼
275 fnressn 6788 . . . . . . . . . . . . 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 4721 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩ = ⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩)
278277sneqd 4488 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → {⟨(1st𝑥), ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥))⟩} = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
279276, 278eqtrd 2831 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)}) = {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩})
280279oveq2d 7037 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}))
281 dprdsubg 18868 . . . . . . . . . . . . 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 18880 . . . . . . . . . . . 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 496 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐺 DProd {⟨(1st𝑥), (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))⟩}) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
286280, 285eqtrd 2831 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ {(1st𝑥)})) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
2874adantr 481 . . . . . . . . . 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 4033 . . . . . . . . . . 11 (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼
290289a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐼 ∖ {(1st𝑥)}) ⊆ 𝐼)
291 disjdif 4339 . . . . . . . . . . 11 ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅
292291a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ({(1st𝑥)} ∩ (𝐼 ∖ {(1st𝑥)})) = ∅)
293287, 288, 169, 290, 292, 1dprdcntz2 18882 . . . . . . . . 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 3931 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
29529adantlr 711 . . . . . . . . . . 11 (((𝜑𝑥𝐴) ∧ 𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
29666, 245, 49, 295, 287, 3, 290dprd2dlem1 18885 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
297 resmpt 5791 . . . . . . . . . . . 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 7032 . . . . . . . . . 10 (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))) = (𝐺 DProd (𝑖 ∈ (𝐼 ∖ {(1st𝑥)}) ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
300296, 299syl6eqr 2849 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))))
301300fveq2d 6547 . . . . . . . 8 ((𝜑𝑥𝐴) → ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = ((Cntz‘𝐺)‘(𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))))
302294, 301sseqtr4d 3933 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
303273, 302sstrd 3903 . . . . . 6 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ ((Cntz‘𝐺)‘(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
304225, 1lsmsubg 18514 . . . . . 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 1364 . . . . 5 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ∈ (SubGrp‘𝐺))
3063mrcsscl 16725 . . . . 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 1364 . . . 4 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ∖ {𝑥}))) ⊆ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
308 sslin 4135 . . . 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)})))))))
31041ffvelrnda 6721 . . . 4 ((𝜑𝑥𝐴) → (𝑆𝑥) ∈ (SubGrp‘𝐺))
311225lsmlub 18523 . . . . . . . . . 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 1364 . . . . . . . . 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 708 . . . . . . . 8 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
314313, 124sseqtr4d 3933 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ⊆ ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)))
315287, 288, 290dprdres 18872 . . . . . . . . . . . 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 495 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → 𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
3173dprdspan 18871 . . . . . . . . . . 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 5461 . . . . . . . . . . . 12 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
320319unieqi 4758 . . . . . . . . . . 11 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})) = ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)}))
321320fveq2i 6546 . . . . . . . . . 10 (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))) = (𝐾 ran ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ (𝐼 ∖ {(1st𝑥)})))
322318, 321syl6eqr 2849 . . . . . . . . 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 2831 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))) = (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)}))))
324 eqimss 3948 . . . . . . . 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 4137 . . . . . . 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 18853 . . . . . 6 ((𝜑𝑥𝐴) → (((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))‘(1st𝑥)) ∩ (𝐾 ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) “ (𝐼 ∖ {(1st𝑥)})))) = {(0g𝐺)})
329327, 328sseqtrd 3932 . . . . 5 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) ⊆ {(0g𝐺)})
330225lsmub2 18517 . . . . . . . . 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 18046 . . . . . . . . 9 ((𝑆𝑥) ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ (𝑆𝑥))
333310, 332syl 17 . . . . . . . 8 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (𝑆𝑥))
334331, 333sseldd 3894 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)))
3352subg0cl 18046 . . . . . . . 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 4096 . . . . . 6 ((𝜑𝑥𝐴) → (0g𝐺) ∈ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
338337snssd 4653 . . . . 5 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))))
339329, 338eqssd 3910 . . . 4 ((𝜑𝑥𝐴) → (((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝑆𝑥)) ∩ (𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)}))))) = {(0g𝐺)})
340 incom 4103 . . . . 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 6547 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆𝑥) = (𝑆‘⟨(1st𝑥), (2nd𝑥)⟩))
34399, 341, 3423eqtr4a 2857 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) = (𝑆𝑥))
344 eqimss2 3949 . . . . . . . . 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 4630 . . . . . . . . . . . . 13 (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ↔ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥))
34711ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → Rel 𝐴)
348 simprl 767 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ (𝐴 ↾ {(1st𝑥)}))
349247, 348sseldi 3891 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝐴)
350347, 349, 74syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
351350fveq2d 6547 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = (𝑆‘⟨(1st𝑦), (2nd𝑦)⟩))
352351, 109syl6eqr 2849 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑦)𝑆(2nd𝑦)))
353350, 348eqeltrrd 2884 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}))
354 fvex 6556 . . . . . . . . . . . . . . . . . . . . . 22 (2nd𝑦) ∈ V
355354opelresi 5747 . . . . . . . . . . . . . . . . . . . . 21 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) ↔ ((1st𝑦) ∈ {(1st𝑥)} ∧ ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝐴))
356355simplbi 498 . . . . . . . . . . . . . . . . . . . 20 (⟨(1st𝑦), (2nd𝑦)⟩ ∈ (𝐴 ↾ {(1st𝑥)}) → (1st𝑦) ∈ {(1st𝑥)})
357353, 356syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) ∈ {(1st𝑥)})
358 elsni 4493 . . . . . . . . . . . . . . . . . . 19 ((1st𝑦) ∈ {(1st𝑥)} → (1st𝑦) = (1st𝑥))
359357, 358syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (1st𝑦) = (1st𝑥))
360359oveq1d 7036 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → ((1st𝑦)𝑆(2nd𝑦)) = ((1st𝑥)𝑆(2nd𝑦)))
361352, 360eqtrd 2831 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) = ((1st𝑥)𝑆(2nd𝑦)))
362348, 230syl6eleq 2893 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})))
363 xp2nd 7583 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ ({(1st𝑥)} × (𝐴 “ {(1st𝑥)})) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
364362, 363syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ∈ (𝐴 “ {(1st𝑥)}))
365 simprr 769 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑦𝑥)
36661adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
367350, 366eqeq12d 2810 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩))
368 fvex 6556 . . . . . . . . . . . . . . . . . . . . . . . 24 (1st𝑦) ∈ V
369368, 354opth 5265 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨(1st𝑦), (2nd𝑦)⟩ = ⟨(1st𝑥), (2nd𝑥)⟩ ↔ ((1st𝑦) = (1st𝑥) ∧ (2nd𝑦) = (2nd𝑥)))
370369baib 536 . . . . . . . . . . . . . . . . . . . . . 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 280 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦 = 𝑥 ↔ (2nd𝑦) = (2nd𝑥)))
373372necon3bid 3028 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑦𝑥 ↔ (2nd𝑦) ≠ (2nd𝑥)))
374365, 373mpbid 233 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (2nd𝑦) ≠ (2nd𝑥))
375 eldifsn 4630 . . . . . . . . . . . . . . . . . 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 7053 . . . . . . . . . . . . . . . . 17 ((1st𝑥)𝑆(2nd𝑦)) ∈ V
378 difss 4033 . . . . . . . . . . . . . . . . . . 19 ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}) ⊆ (𝐴 “ {(1st𝑥)})
379 resmpt 5791 . . . . . . . . . . . . . . . . . . 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 7029 . . . . . . . . . . . . . . . . . 18 (𝑗 = (2nd𝑦) → ((1st𝑥)𝑆𝑗) = ((1st𝑥)𝑆(2nd𝑦)))
382380, 381elrnmpt1s 5716 . . . . . . . . . . . . . . . . 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 2883 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
385 df-ima 5461 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) = ran ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) ↾ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))
386384, 385syl6eleqr 2894 . . . . . . . . . . . . . 14 (((𝜑𝑥𝐴) ∧ (𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥)) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
387386ex 413 . . . . . . . . . . . . 13 ((𝜑𝑥𝐴) → ((𝑦 ∈ (𝐴 ↾ {(1st𝑥)}) ∧ 𝑦𝑥) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
388346, 387syl5bi 243 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) → (𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
389388ralrimiv 3148 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → ∀𝑦 ∈ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})(𝑆𝑦) ∈ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
390231, 250sstrid 3904 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}) ⊆ dom 𝑆)
391 funimass4 6603 . . . . . . . . . . . 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 258 . . . . . . . . . 10 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
394393unissd 4773 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})) ⊆ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))
395 imassrn 5822 . . . . . . . . . . 11 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ ran (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))
396395, 267sstrid 3904 . . . . . . . . . 10 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺))
397 sspwuni 4925 . . . . . . . . . 10 (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ 𝒫 (Base‘𝐺) ↔ ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
398396, 397sylib 219 . . . . . . . . 9 ((𝜑𝑥𝐴) → ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})) ⊆ (Base‘𝐺))
399166, 3, 394, 398mrcssd 16729 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ⊆ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)}))))
400 ss2in 4137 . . . . . . . 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 18853 . . . . . . 7 ((𝜑𝑥𝐴) → (((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))‘(2nd𝑥)) ∩ (𝐾 ((𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)) “ ((𝐴 “ {(1st𝑥)}) ∖ {(2nd𝑥)})))) = {(0g𝐺)})
404401, 403sseqtrd 3932 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) ⊆ {(0g𝐺)})
4052subg0cl 18046 . . . . . . . . 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 4096 . . . . . . 7 ((𝜑𝑥𝐴) → (0g𝐺) ∈ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
408407snssd 4653 . . . . . 6 ((𝜑𝑥𝐴) → {(0g𝐺)} ⊆ ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))))
409404, 408eqssd 3910 . . . . 5 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))) = {(0g𝐺)})
410340, 409syl5eq 2843 . . . 4 ((𝜑𝑥𝐴) → ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥}))) ∩ (𝑆𝑥)) = {(0g𝐺)})
411225, 222, 310, 224, 2, 339, 410lsmdisj2 18540 . . 3 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ ((𝐾 (𝑆 “ ((𝐴 ↾ {(1st𝑥)}) ∖ {𝑥})))(LSSum‘𝐺)(𝐾 (𝑆 “ (𝐴 ↾ (𝐼 ∖ {(1st𝑥)})))))) = {(0g𝐺)})
412309, 411sseqtrd 3932 . 2 ((𝜑𝑥𝐴) → ((𝑆𝑥) ∩ (𝐾 (𝑆 “ (𝐴 ∖ {𝑥})))) ⊆ {(0g𝐺)})
4131, 2, 3, 6, 40, 41, 162, 412dmdprdd 18843 1 (𝜑𝐺dom DProd 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  Vcvv 3437  cdif 3860  cun 3861  cin 3862  wss 3863  c0 4215  𝒫 cpw 4457  {csn 4476  cop 4482   cuni 4749   ciun 4829   class class class wbr 4966  cmpt 5045   × cxp 5446  dom cdm 5448  ran crn 5449  cres 5450  cima 5451  Rel wrel 5453  Fun wfun 6224   Fn wfn 6225  wf 6226  cfv 6230  (class class class)co 7021  1st c1st 7548  2nd c2nd 7549  Basecbs 16317  0gc0g 16547  Moorecmre 16687  mrClscmrc 16688  ACScacs 16690  Grpcgrp 17866  SubGrpcsubg 18032  Cntzccntz 18191  LSSumclsm 18494   DProd cdprd 18837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-int 4787  df-iun 4831  df-iin 4832  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-se 5408  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-isom 6239  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-of 7272  df-om 7442  df-1st 7550  df-2nd 7551  df-supp 7687  df-tpos 7748  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-1o 7958  df-oadd 7962  df-er 8144  df-map 8263  df-ixp 8316  df-en 8363  df-dom 8364  df-sdom 8365  df-fin 8366  df-fsupp 8685  df-oi 8825  df-card 9219  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-n0 11751  df-z 11835  df-uz 12099  df-fz 12748  df-fzo 12889  df-seq 13225  df-hash 13546  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-0g 16549  df-gsum 16550  df-mre 16691  df-mrc 16692  df-acs 16694  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-mhm 17779  df-submnd 17780  df-grp 17869  df-minusg 17870  df-sbg 17871  df-mulg 17987  df-subg 18035  df-ghm 18102  df-gim 18145  df-cntz 18193  df-oppg 18220  df-lsm 18496  df-cmn 18640  df-dprd 18839
This theorem is referenced by:  dprd2db  18887  dprd2d2  18888
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