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Theorem dprd2dlem1 18903
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 18867 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 eqid 2772 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
54subgacs 18088 . . . . 5 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6 acsmre 16771 . . . . 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 6341 . . . . . 6 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11 funiunfv 6826 . . . . . 6 (Fun 𝑆 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
129, 10, 113syl 18 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
13 resss 5717 . . . . . . . . . 10 (𝐴𝐶) ⊆ 𝐴
1413sseli 3850 . . . . . . . . 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 18902 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
1914, 18sylan2 583 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
20 1st2nd 7543 . . . . . . . . . . . . 13 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2115, 14, 20syl2an 586 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
22 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
2321, 22eqeltrrd 2861 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴𝐶)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶))
24 fvex 6506 . . . . . . . . . . . . 13 (2nd𝑥) ∈ V
2524opelresi 5696 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) ↔ ((1st𝑥) ∈ 𝐶 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴))
2625simplbi 490 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) → (1st𝑥) ∈ 𝐶)
2723, 26syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴𝐶)) → (1st𝑥) ∈ 𝐶)
28 ovex 7002 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ V
29 eqid 2772 . . . . . . . . . . 11 (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30 sneq 4445 . . . . . . . . . . . . . 14 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
3130imaeq2d 5764 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
32 oveq1 6977 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
3331, 32mpteq12dv 5006 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
3433oveq2d 6986 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
3529, 34elrnmpt1s 5665 . . . . . . . . . 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 577 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
37 elssuni 4735 . . . . . . . . 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 3864 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4039ralrimiva 3126 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41 iunss 4829 . . . . . 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 226 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4312, 42eqsstr3d 3892 . . . 4 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44 dprd2d.6 . . . . . . . . . . . 12 (𝜑𝐶𝐼)
4544sselda 3854 . . . . . . . . . . 11 ((𝜑𝑖𝐶) → 𝑖𝐼)
4645, 17syldan 582 . . . . . . . . . 10 ((𝜑𝑖𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47 ovex 7002 . . . . . . . . . . . 12 (𝑖𝑆𝑗) ∈ V
48 eqid 2772 . . . . . . . . . . . 12 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
4947, 48dmmpti 6316 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
5049a1i 11 . . . . . . . . . 10 ((𝜑𝑖𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51 imassrn 5775 . . . . . . . . . . . . . 14 (𝑆 “ (𝐴𝐶)) ⊆ ran 𝑆
529frnd 6345 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53 mresspw 16711 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
547, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5552, 54sstrd 3864 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5651, 55syl5ss 3865 . . . . . . . . . . . . 13 (𝜑 → (𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺))
57 sspwuni 4882 . . . . . . . . . . . . 13 ((𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
5856, 57sylib 210 . . . . . . . . . . . 12 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
598mrccl 16730 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
607, 58, 59syl2anc 576 . . . . . . . . . . 11 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
6160adantr 473 . . . . . . . . . 10 ((𝜑𝑖𝐶) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
62 oveq2 6978 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
6362, 48, 47fvmpt3i 6594 . . . . . . . . . . . 12 (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
6463adantl 474 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65 df-ov 6973 . . . . . . . . . . . . . 14 (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
669ffnd 6339 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Fn 𝐴)
6766ad2antrr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
6813a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴𝐶) ⊆ 𝐴)
69 simplr 756 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐶)
70 elrelimasn 5787 . . . . . . . . . . . . . . . . . . . 20 (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7115, 70syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7271adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7372biimpa 469 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74 df-br 4924 . . . . . . . . . . . . . . . . 17 (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
7573, 74sylib 210 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76 vex 3412 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
7776opelresi 5696 . . . . . . . . . . . . . . . 16 (⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶) ↔ (𝑖𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
7869, 75, 77sylanbrc 575 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶))
79 fnfvima 6814 . . . . . . . . . . . . . . 15 ((𝑆 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8067, 68, 78, 79syl3anc 1351 . . . . . . . . . . . . . 14 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8165, 80syl5eqel 2864 . . . . . . . . . . . . 13 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)))
82 elssuni 4735 . . . . . . . . . . . . 13 ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
8381, 82syl 17 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
847, 8, 58mrcssidd 16744 . . . . . . . . . . . . 13 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8584ad2antrr 713 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8683, 85sstrd 3864 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8764, 86eqsstrd 3891 . . . . . . . . . 10 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8846, 50, 61, 87dprdlub 18888 . . . . . . . . 9 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
89 ovex 7002 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
9089elpw 4422 . . . . . . . . 9 ((𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9188, 90sylibr 226 . . . . . . . 8 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9291fmpttd 6696 . . . . . . 7 (𝜑 → (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9392frnd 6345 . . . . . 6 (𝜑 → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
94 sspwuni 4882 . . . . . 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 210 . . . . 5 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
967, 8mrcssvd 16742 . . . . 5 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (Base‘𝐺))
9795, 96sstrd 3864 . . . 4 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
987, 8, 43, 97mrcssd 16743 . . 3 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
998mrcsscl 16739 . . . 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 1351 . . 3 (𝜑 → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
10198, 100eqssd 3871 . 2 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102 eqid 2772 . . . . . . . 8 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
10389, 102dmmpti 6316 . . . . . . 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 18890 . . . . 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 487 . . . 4 (𝜑𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
10744resmptd 5747 . . . 4 (𝜑 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108106, 107breqtrd 4949 . . 3 (𝜑𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
1098dprdspan 18889 . . 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 2811 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 198  wa 387   = wceq 1507  wcel 2048  wral 3082  Vcvv 3409  wss 3825  𝒫 cpw 4416  {csn 4435  cop 4441   cuni 4706   ciun 4786   class class class wbr 4923  cmpt 5002  dom cdm 5400  ran crn 5401  cres 5402  cima 5403  Rel wrel 5405  Fun wfun 6176   Fn wfn 6177  wf 6178  cfv 6182  (class class class)co 6970  1st c1st 7492  2nd c2nd 7493  Basecbs 16329  Moorecmre 16701  mrClscmrc 16702  ACScacs 16704  Grpcgrp 17881  SubGrpcsubg 18047   DProd cdprd 18855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7494  df-2nd 7495  df-supp 7627  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-oadd 7901  df-er 8081  df-map 8200  df-ixp 8252  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-fsupp 8621  df-oi 8761  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-n0 11701  df-z 11787  df-uz 12052  df-fz 12702  df-fzo 12843  df-seq 13178  df-hash 13499  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-0g 16561  df-gsum 16562  df-mre 16705  df-mrc 16706  df-acs 16708  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-mhm 17793  df-submnd 17794  df-grp 17884  df-minusg 17885  df-sbg 17886  df-mulg 18002  df-subg 18050  df-ghm 18117  df-gim 18160  df-cntz 18208  df-oppg 18235  df-cmn 18658  df-dprd 18857
This theorem is referenced by:  dprd2da  18904  dprd2db  18905
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