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Theorem dprd2dlem1 19731
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))
dprd2d.6 (𝜑𝐶𝐼)
Assertion
Ref Expression
dprd2dlem1 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
Distinct variable groups:   𝑖,𝑗,𝐴   𝐶,𝑖   𝑖,𝐺,𝑗   𝑖,𝐼   𝑖,𝐾   𝜑,𝑖,𝑗   𝑆,𝑖,𝑗
Allowed substitution hints:   𝐶(𝑗)   𝐼(𝑗)   𝐾(𝑗)

Proof of Theorem dprd2dlem1
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprd2d.5 . . . . . 6 (𝜑𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
2 dprdgrp 19695 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 eqid 2736 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
54subgacs 18877 . . . . 5 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6 acsmre 17450 . . . . 5 ((SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)) → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
73, 5, 63syl 18 . . . 4 (𝜑 → (SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)))
8 dprd2d.k . . . 4 𝐾 = (mrCls‘(SubGrp‘𝐺))
9 dprd2d.2 . . . . . 6 (𝜑𝑆:𝐴⟶(SubGrp‘𝐺))
10 ffun 6648 . . . . . 6 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11 funiunfv 7171 . . . . . 6 (Fun 𝑆 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
129, 10, 113syl 18 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
13 resss 5942 . . . . . . . . . 10 (𝐴𝐶) ⊆ 𝐴
1413sseli 3927 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐶) → 𝑥𝐴)
15 dprd2d.1 . . . . . . . . . 10 (𝜑 → Rel 𝐴)
16 dprd2d.3 . . . . . . . . . 10 (𝜑 → dom 𝐴𝐼)
17 dprd2d.4 . . . . . . . . . 10 ((𝜑𝑖𝐼) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
1815, 9, 16, 17, 1, 8dprd2dlem2 19730 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
1914, 18sylan2 593 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
20 1st2nd 7940 . . . . . . . . . . . . 13 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2115, 14, 20syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
22 simpr 485 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
2321, 22eqeltrrd 2838 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴𝐶)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶))
24 fvex 6832 . . . . . . . . . . . . 13 (2nd𝑥) ∈ V
2524opelresi 5925 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) ↔ ((1st𝑥) ∈ 𝐶 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴))
2625simplbi 498 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) → (1st𝑥) ∈ 𝐶)
2723, 26syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴𝐶)) → (1st𝑥) ∈ 𝐶)
28 ovex 7362 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ V
29 eqid 2736 . . . . . . . . . . 11 (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30 sneq 4582 . . . . . . . . . . . . . 14 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
3130imaeq2d 5993 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
32 oveq1 7336 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
3331, 32mpteq12dv 5180 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
3433oveq2d 7345 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
3529, 34elrnmpt1s 5892 . . . . . . . . . 10 (((1st𝑥) ∈ 𝐶 ∧ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ V) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
3627, 28, 35sylancl 586 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
37 elssuni 4884 . . . . . . . . 9 ((𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
3836, 37syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
3919, 38sstrd 3941 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4039ralrimiva 3139 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41 iunss 4989 . . . . . 6 ( 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↔ ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4240, 41sylibr 233 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4312, 42eqsstrrd 3970 . . . 4 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44 dprd2d.6 . . . . . . . . . . . 12 (𝜑𝐶𝐼)
4544sselda 3931 . . . . . . . . . . 11 ((𝜑𝑖𝐶) → 𝑖𝐼)
4645, 17syldan 591 . . . . . . . . . 10 ((𝜑𝑖𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47 ovex 7362 . . . . . . . . . . . 12 (𝑖𝑆𝑗) ∈ V
48 eqid 2736 . . . . . . . . . . . 12 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
4947, 48dmmpti 6622 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
5049a1i 11 . . . . . . . . . 10 ((𝜑𝑖𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51 imassrn 6004 . . . . . . . . . . . . . 14 (𝑆 “ (𝐴𝐶)) ⊆ ran 𝑆
529frnd 6653 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53 mresspw 17390 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
547, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5552, 54sstrd 3941 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5651, 55sstrid 3942 . . . . . . . . . . . . 13 (𝜑 → (𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺))
57 sspwuni 5044 . . . . . . . . . . . . 13 ((𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
5856, 57sylib 217 . . . . . . . . . . . 12 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
598mrccl 17409 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
607, 58, 59syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
6160adantr 481 . . . . . . . . . 10 ((𝜑𝑖𝐶) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
62 oveq2 7337 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
6362, 48, 47fvmpt3i 6930 . . . . . . . . . . . 12 (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
6463adantl 482 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65 df-ov 7332 . . . . . . . . . . . . . 14 (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
669ffnd 6646 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Fn 𝐴)
6766ad2antrr 723 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
6813a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴𝐶) ⊆ 𝐴)
69 simplr 766 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐶)
70 elrelimasn 6017 . . . . . . . . . . . . . . . . . . . 20 (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7115, 70syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7271adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7372biimpa 477 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74 df-br 5090 . . . . . . . . . . . . . . . . 17 (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
7573, 74sylib 217 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76 vex 3445 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
7776opelresi 5925 . . . . . . . . . . . . . . . 16 (⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶) ↔ (𝑖𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
7869, 75, 77sylanbrc 583 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶))
79 fnfvima 7159 . . . . . . . . . . . . . . 15 ((𝑆 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8067, 68, 78, 79syl3anc 1370 . . . . . . . . . . . . . 14 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8165, 80eqeltrid 2841 . . . . . . . . . . . . 13 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)))
82 elssuni 4884 . . . . . . . . . . . . 13 ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
8381, 82syl 17 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
847, 8, 58mrcssidd 17423 . . . . . . . . . . . . 13 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8584ad2antrr 723 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8683, 85sstrd 3941 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8764, 86eqsstrd 3969 . . . . . . . . . 10 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8846, 50, 61, 87dprdlub 19716 . . . . . . . . 9 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
89 ovex 7362 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
9089elpw 4550 . . . . . . . . 9 ((𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9188, 90sylibr 233 . . . . . . . 8 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9291fmpttd 7039 . . . . . . 7 (𝜑 → (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9392frnd 6653 . . . . . 6 (𝜑 → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
94 sspwuni 5044 . . . . . 6 (ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9593, 94sylib 217 . . . . 5 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
967, 8mrcssvd 17421 . . . . 5 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (Base‘𝐺))
9795, 96sstrd 3941 . . . 4 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
987, 8, 43, 97mrcssd 17422 . . 3 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
998mrcsscl 17418 . . . 4 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))) ∧ (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺)) → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
1007, 95, 60, 99syl3anc 1370 . . 3 (𝜑 → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
10198, 100eqssd 3948 . 2 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102 eqid 2736 . . . . . . . 8 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
10389, 102dmmpti 6622 . . . . . . 7 dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
104103a1i 11 . . . . . 6 (𝜑 → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
1051, 104, 44dprdres 19718 . . . . 5 (𝜑 → (𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) ∧ (𝐺 DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶)) ⊆ (𝐺 DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))))
106105simpld 495 . . . 4 (𝜑𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
10744resmptd 5974 . . . 4 (𝜑 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108106, 107breqtrd 5115 . . 3 (𝜑𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
1098dprdspan 19717 . . 3 (𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
110108, 109syl 17 . 2 (𝜑 → (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
111101, 110eqtr4d 2779 1 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wral 3061  Vcvv 3441  wss 3897  𝒫 cpw 4546  {csn 4572  cop 4578   cuni 4851   ciun 4938   class class class wbr 5089  cmpt 5172  dom cdm 5614  ran crn 5615  cres 5616  cima 5617  Rel wrel 5619  Fun wfun 6467   Fn wfn 6468  wf 6469  cfv 6473  (class class class)co 7329  1st c1st 7889  2nd c2nd 7890  Basecbs 17001  Moorecmre 17380  mrClscmrc 17381  ACScacs 17383  Grpcgrp 18665  SubGrpcsubg 18837   DProd cdprd 19683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-int 4894  df-iun 4940  df-iin 4941  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-se 5570  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-isom 6482  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-of 7587  df-om 7773  df-1st 7891  df-2nd 7892  df-supp 8040  df-tpos 8104  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-er 8561  df-map 8680  df-ixp 8749  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-fsupp 9219  df-oi 9359  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-2 12129  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476  df-seq 13815  df-hash 14138  df-sets 16954  df-slot 16972  df-ndx 16984  df-base 17002  df-ress 17031  df-plusg 17064  df-0g 17241  df-gsum 17242  df-mre 17384  df-mrc 17385  df-acs 17387  df-mgm 18415  df-sgrp 18464  df-mnd 18475  df-mhm 18519  df-submnd 18520  df-grp 18668  df-minusg 18669  df-sbg 18670  df-mulg 18789  df-subg 18840  df-ghm 18920  df-gim 18963  df-cntz 19011  df-oppg 19038  df-cmn 19475  df-dprd 19685
This theorem is referenced by:  dprd2da  19732  dprd2db  19733
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