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Theorem dprd2dlem1 18707
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 18671 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 eqid 2765 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
54subgacs 17893 . . . . 5 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6 acsmre 16578 . . . . 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 6226 . . . . . 6 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11 funiunfv 6698 . . . . . 6 (Fun 𝑆 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
129, 10, 113syl 18 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
13 resss 5597 . . . . . . . . . 10 (𝐴𝐶) ⊆ 𝐴
1413sseli 3757 . . . . . . . . 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 18706 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
1914, 18sylan2 586 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
20 1st2nd 7414 . . . . . . . . . . . . 13 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2115, 14, 20syl2an 589 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
22 simpr 477 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
2321, 22eqeltrrd 2845 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴𝐶)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶))
24 fvex 6388 . . . . . . . . . . . . 13 (2nd𝑥) ∈ V
2524opelresi 5573 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) ↔ ((1st𝑥) ∈ 𝐶 ∧ ⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴))
2625simplbi 491 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) → (1st𝑥) ∈ 𝐶)
2723, 26syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴𝐶)) → (1st𝑥) ∈ 𝐶)
28 ovex 6874 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ V
29 eqid 2765 . . . . . . . . . . 11 (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30 sneq 4344 . . . . . . . . . . . . . 14 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
3130imaeq2d 5648 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
32 oveq1 6849 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
3331, 32mpteq12dv 4892 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
3433oveq2d 6858 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
3529, 34elrnmpt1s 5542 . . . . . . . . . 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 580 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
37 elssuni 4625 . . . . . . . . 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 3771 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4039ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41 iunss 4717 . . . . . 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 225 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4312, 42eqsstr3d 3800 . . . 4 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44 dprd2d.6 . . . . . . . . . . . 12 (𝜑𝐶𝐼)
4544sselda 3761 . . . . . . . . . . 11 ((𝜑𝑖𝐶) → 𝑖𝐼)
4645, 17syldan 585 . . . . . . . . . 10 ((𝜑𝑖𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47 ovex 6874 . . . . . . . . . . . 12 (𝑖𝑆𝑗) ∈ V
48 eqid 2765 . . . . . . . . . . . 12 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
4947, 48dmmpti 6201 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
5049a1i 11 . . . . . . . . . 10 ((𝜑𝑖𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51 imassrn 5659 . . . . . . . . . . . . . 14 (𝑆 “ (𝐴𝐶)) ⊆ ran 𝑆
529frnd 6230 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
53 mresspw 16518 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
547, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5552, 54sstrd 3771 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5651, 55syl5ss 3772 . . . . . . . . . . . . 13 (𝜑 → (𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺))
57 sspwuni 4768 . . . . . . . . . . . . 13 ((𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
5856, 57sylib 209 . . . . . . . . . . . 12 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
598mrccl 16537 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
607, 58, 59syl2anc 579 . . . . . . . . . . 11 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
6160adantr 472 . . . . . . . . . 10 ((𝜑𝑖𝐶) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
62 oveq2 6850 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
6362, 48, 47fvmpt3i 6476 . . . . . . . . . . . 12 (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
6463adantl 473 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
65 df-ov 6845 . . . . . . . . . . . . . 14 (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
669ffnd 6224 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Fn 𝐴)
6766ad2antrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
6813a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴𝐶) ⊆ 𝐴)
69 simplr 785 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐶)
70 elrelimasn 5671 . . . . . . . . . . . . . . . . . . . 20 (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7115, 70syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7271adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7372biimpa 468 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
74 df-br 4810 . . . . . . . . . . . . . . . . 17 (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
7573, 74sylib 209 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
76 vex 3353 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
7776opelresi 5573 . . . . . . . . . . . . . . . 16 (⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶) ↔ (𝑖𝐶 ∧ ⟨𝑖, 𝑘⟩ ∈ 𝐴))
7869, 75, 77sylanbrc 578 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶))
79 fnfvima 6689 . . . . . . . . . . . . . . 15 ((𝑆 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8067, 68, 78, 79syl3anc 1490 . . . . . . . . . . . . . 14 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8165, 80syl5eqel 2848 . . . . . . . . . . . . 13 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)))
82 elssuni 4625 . . . . . . . . . . . . 13 ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
8381, 82syl 17 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
847, 8, 58mrcssidd 16551 . . . . . . . . . . . . 13 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8584ad2antrr 717 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8683, 85sstrd 3771 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8764, 86eqsstrd 3799 . . . . . . . . . 10 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8846, 50, 61, 87dprdlub 18692 . . . . . . . . 9 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
89 ovex 6874 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
9089elpw 4321 . . . . . . . . 9 ((𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9188, 90sylibr 225 . . . . . . . 8 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9291fmpttd 6575 . . . . . . 7 (𝜑 → (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9392frnd 6230 . . . . . 6 (𝜑 → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
94 sspwuni 4768 . . . . . 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 209 . . . . 5 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
967, 8mrcssvd 16549 . . . . 5 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (Base‘𝐺))
9795, 96sstrd 3771 . . . 4 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
987, 8, 43, 97mrcssd 16550 . . 3 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
998mrcsscl 16546 . . . 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 1490 . . 3 (𝜑 → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
10198, 100eqssd 3778 . 2 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
102 eqid 2765 . . . . . . . 8 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
10389, 102dmmpti 6201 . . . . . . 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 18694 . . . . 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 488 . . . 4 (𝜑𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
10744resmptd 5629 . . . 4 (𝜑 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
108106, 107breqtrd 4835 . . 3 (𝜑𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
1098dprdspan 18693 . . 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 2802 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 197  wa 384   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  wss 3732  𝒫 cpw 4315  {csn 4334  cop 4340   cuni 4594   ciun 4676   class class class wbr 4809  cmpt 4888  dom cdm 5277  ran crn 5278  cres 5279  cima 5280  Rel wrel 5282  Fun wfun 6062   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  1st c1st 7364  2nd c2nd 7365  Basecbs 16130  Moorecmre 16508  mrClscmrc 16509  ACScacs 16511  Grpcgrp 17689  SubGrpcsubg 17852   DProd cdprd 18659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-0g 16368  df-gsum 16369  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-gim 17965  df-cntz 18013  df-oppg 18039  df-cmn 18461  df-dprd 18661
This theorem is referenced by:  dprd2da  18708  dprd2db  18709
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