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Theorem dprd2dlem1 19820
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 19784 . . . . . 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 18963 . . . . 5 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6 acsmre 17532 . . . . 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 6671 . . . . . 6 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11 funiunfv 7195 . . . . . 6 (Fun 𝑆 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
129, 10, 113syl 18 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
13 resss 5962 . . . . . . . . . 10 (𝐴𝐶) ⊆ 𝐴
1413sseli 3940 . . . . . . . . 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 19819 . . . . . . . . 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 7971 . . . . . . . . . . . . 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 2839 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴𝐶)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶))
24 fvex 6855 . . . . . . . . . . . . 13 (2nd𝑥) ∈ V
2524opelresi 5945 . . . . . . . . . . . 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 7390 . . . . . . . . . 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 4596 . . . . . . . . . . . . . 14 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
3130imaeq2d 6013 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
32 oveq1 7364 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
3331, 32mpteq12dv 5196 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
3433oveq2d 7373 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
3529, 34elrnmpt1s 5912 . . . . . . . . . 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 4898 . . . . . . . . 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 3954 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4039ralrimiva 3143 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41 iunss 5005 . . . . . 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 3983 . . . 4 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44 dprd2d.6 . . . . . . . . . . . 12 (𝜑𝐶𝐼)
4544sselda 3944 . . . . . . . . . . 11 ((𝜑𝑖𝐶) → 𝑖𝐼)
4645, 17syldan 591 . . . . . . . . . 10 ((𝜑𝑖𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47 ovex 7390 . . . . . . . . . . . 12 (𝑖𝑆𝑗) ∈ V
48 eqid 2736 . . . . . . . . . . . 12 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
4947, 48dmmpti 6645 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
5049a1i 11 . . . . . . . . . 10 ((𝜑𝑖𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51 imassrn 6024 . . . . . . . . . . . . . 14 (𝑆 “ (𝐴𝐶)) ⊆ ran 𝑆
529frnd 6676 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53 mresspw 17472 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
547, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5552, 54sstrd 3954 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5651, 55sstrid 3955 . . . . . . . . . . . . 13 (𝜑 → (𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺))
57 sspwuni 5060 . . . . . . . . . . . . 13 ((𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
5856, 57sylib 217 . . . . . . . . . . . 12 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
598mrccl 17491 . . . . . . . . . . . 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 7365 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
6362, 48, 47fvmpt3i 6953 . . . . . . . . . . . 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 7360 . . . . . . . . . . . . . 14 (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
669ffnd 6669 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Fn 𝐴)
6766ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
6813a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴𝐶) ⊆ 𝐴)
69 simplr 767 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐶)
70 elrelimasn 6037 . . . . . . . . . . . . . . . . . . . 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 5106 . . . . . . . . . . . . . . . . 17 (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
7573, 74sylib 217 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76 vex 3449 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
7776opelresi 5945 . . . . . . . . . . . . . . . 16 (⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶) ↔ (𝑖𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
7869, 75, 77sylanbrc 583 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶))
79 fnfvima 7183 . . . . . . . . . . . . . . 15 ((𝑆 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8067, 68, 78, 79syl3anc 1371 . . . . . . . . . . . . . 14 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8165, 80eqeltrid 2842 . . . . . . . . . . . . 13 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)))
82 elssuni 4898 . . . . . . . . . . . . 13 ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
8381, 82syl 17 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
847, 8, 58mrcssidd 17505 . . . . . . . . . . . . 13 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8584ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8683, 85sstrd 3954 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8764, 86eqsstrd 3982 . . . . . . . . . 10 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8846, 50, 61, 87dprdlub 19805 . . . . . . . . 9 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
89 ovex 7390 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
9089elpw 4564 . . . . . . . . 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 7063 . . . . . . 7 (𝜑 → (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9392frnd 6676 . . . . . 6 (𝜑 → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
94 sspwuni 5060 . . . . . 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 17503 . . . . 5 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (Base‘𝐺))
9795, 96sstrd 3954 . . . 4 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
987, 8, 43, 97mrcssd 17504 . . 3 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
998mrcsscl 17500 . . . 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 1371 . . 3 (𝜑 → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
10198, 100eqssd 3961 . 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 6645 . . . . . . 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 19807 . . . . 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 5994 . . . 4 (𝜑 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108106, 107breqtrd 5131 . . 3 (𝜑𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
1098dprdspan 19806 . . 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 1541  wcel 2106  wral 3064  Vcvv 3445  wss 3910  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Rel wrel 5638  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Moorecmre 17462  mrClscmrc 17463  ACScacs 17465  Grpcgrp 18748  SubGrpcsubg 18922   DProd cdprd 19772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-gim 19049  df-cntz 19097  df-oppg 19124  df-cmn 19564  df-dprd 19774
This theorem is referenced by:  dprd2da  19821  dprd2db  19822
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