Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprd2dlem1 Structured version   Visualization version   GIF version

Theorem dprd2dlem1 18263
 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 18227 . . . . . 6 (𝐺dom DProd (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 eqid 2610 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
54subgacs 17452 . . . . 5 (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ (ACS‘(Base‘𝐺)))
6 acsmre 16136 . . . . 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 5961 . . . . . 6 (𝑆:𝐴⟶(SubGrp‘𝐺) → Fun 𝑆)
11 funiunfv 6410 . . . . . 6 (Fun 𝑆 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
129, 10, 113syl 18 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) = (𝑆 “ (𝐴𝐶)))
13 resss 5342 . . . . . . . . . 10 (𝐴𝐶) ⊆ 𝐴
1413sseli 3564 . . . . . . . . 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 18262 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
1914, 18sylan2 490 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
20 1st2nd 7105 . . . . . . . . . . . . 13 ((Rel 𝐴𝑥𝐴) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
2115, 14, 20syl2an 493 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 = ⟨(1st𝑥), (2nd𝑥)⟩)
22 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
2321, 22eqeltrrd 2689 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝐴𝐶)) → ⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶))
24 fvex 6113 . . . . . . . . . . . . 13 (2nd𝑥) ∈ V
2524opelres 5322 . . . . . . . . . . . 12 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) ↔ (⟨(1st𝑥), (2nd𝑥)⟩ ∈ 𝐴 ∧ (1st𝑥) ∈ 𝐶))
2625simprbi 479 . . . . . . . . . . 11 (⟨(1st𝑥), (2nd𝑥)⟩ ∈ (𝐴𝐶) → (1st𝑥) ∈ 𝐶)
2723, 26syl 17 . . . . . . . . . 10 ((𝜑𝑥 ∈ (𝐴𝐶)) → (1st𝑥) ∈ 𝐶)
28 ovex 6577 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ V
29 eqid 2610 . . . . . . . . . . 11 (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
30 sneq 4135 . . . . . . . . . . . . . 14 (𝑖 = (1st𝑥) → {𝑖} = {(1st𝑥)})
3130imaeq2d 5385 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝐴 “ {𝑖}) = (𝐴 “ {(1st𝑥)}))
32 oveq1 6556 . . . . . . . . . . . . 13 (𝑖 = (1st𝑥) → (𝑖𝑆𝑗) = ((1st𝑥)𝑆𝑗))
3331, 32mpteq12dv 4663 . . . . . . . . . . . 12 (𝑖 = (1st𝑥) → (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗)))
3433oveq2d 6565 . . . . . . . . . . 11 (𝑖 = (1st𝑥) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) = (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))))
3529, 34elrnmpt1s 5294 . . . . . . . . . 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 693 . . . . . . . . 9 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {(1st𝑥)}) ↦ ((1st𝑥)𝑆𝑗))) ∈ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
37 elssuni 4403 . . . . . . . . 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 3578 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴𝐶)) → (𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4039ralrimiva 2949 . . . . . 6 (𝜑 → ∀𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
41 iunss 4497 . . . . . 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 223 . . . . 5 (𝜑 𝑥 ∈ (𝐴𝐶)(𝑆𝑥) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
4312, 42eqsstr3d 3603 . . . 4 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
44 dprd2d.6 . . . . . . . . . . . 12 (𝜑𝐶𝐼)
4544sselda 3568 . . . . . . . . . . 11 ((𝜑𝑖𝐶) → 𝑖𝐼)
4645, 17syldan 486 . . . . . . . . . 10 ((𝜑𝑖𝐶) → 𝐺dom DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))
47 ovex 6577 . . . . . . . . . . . 12 (𝑖𝑆𝑗) ∈ V
48 eqid 2610 . . . . . . . . . . . 12 (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))
4947, 48dmmpti 5936 . . . . . . . . . . 11 dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖})
5049a1i 11 . . . . . . . . . 10 ((𝜑𝑖𝐶) → dom (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)) = (𝐴 “ {𝑖}))
51 imassrn 5396 . . . . . . . . . . . . . 14 (𝑆 “ (𝐴𝐶)) ⊆ ran 𝑆
52 frn 5966 . . . . . . . . . . . . . . . 16 (𝑆:𝐴⟶(SubGrp‘𝐺) → ran 𝑆 ⊆ (SubGrp‘𝐺))
539, 52syl 17 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝑆 ⊆ (SubGrp‘𝐺))
54 mresspw 16075 . . . . . . . . . . . . . . . 16 ((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
557, 54syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (SubGrp‘𝐺) ⊆ 𝒫 (Base‘𝐺))
5653, 55sstrd 3578 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑆 ⊆ 𝒫 (Base‘𝐺))
5751, 56syl5ss 3579 . . . . . . . . . . . . 13 (𝜑 → (𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺))
58 sspwuni 4547 . . . . . . . . . . . . 13 ((𝑆 “ (𝐴𝐶)) ⊆ 𝒫 (Base‘𝐺) ↔ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
5957, 58sylib 207 . . . . . . . . . . . 12 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺))
608mrccl 16094 . . . . . . . . . . . 12 (((SubGrp‘𝐺) ∈ (Moore‘(Base‘𝐺)) ∧ (𝑆 “ (𝐴𝐶)) ⊆ (Base‘𝐺)) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
617, 59, 60syl2anc 691 . . . . . . . . . . 11 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
6261adantr 480 . . . . . . . . . 10 ((𝜑𝑖𝐶) → (𝐾 (𝑆 “ (𝐴𝐶))) ∈ (SubGrp‘𝐺))
63 oveq2 6557 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑖𝑆𝑗) = (𝑖𝑆𝑘))
6463, 48, 47fvmpt3i 6196 . . . . . . . . . . . 12 (𝑘 ∈ (𝐴 “ {𝑖}) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
6564adantl 481 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) = (𝑖𝑆𝑘))
66 df-ov 6552 . . . . . . . . . . . . . 14 (𝑖𝑆𝑘) = (𝑆‘⟨𝑖, 𝑘⟩)
67 ffn 5958 . . . . . . . . . . . . . . . . 17 (𝑆:𝐴⟶(SubGrp‘𝐺) → 𝑆 Fn 𝐴)
689, 67syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑆 Fn 𝐴)
6968ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑆 Fn 𝐴)
7013a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝐴𝐶) ⊆ 𝐴)
71 elrelimasn 5408 . . . . . . . . . . . . . . . . . . . 20 (Rel 𝐴 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7215, 71syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7372adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖𝐶) → (𝑘 ∈ (𝐴 “ {𝑖}) ↔ 𝑖𝐴𝑘))
7473biimpa 500 . . . . . . . . . . . . . . . . 17 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐴𝑘)
75 df-br 4584 . . . . . . . . . . . . . . . . 17 (𝑖𝐴𝑘 ↔ ⟨𝑖, 𝑘⟩ ∈ 𝐴)
7674, 75sylib 207 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ 𝐴)
77 simplr 788 . . . . . . . . . . . . . . . 16 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → 𝑖𝐶)
78 vex 3176 . . . . . . . . . . . . . . . . 17 𝑘 ∈ V
7978opelres 5322 . . . . . . . . . . . . . . . 16 (⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶) ↔ (⟨𝑖, 𝑘⟩ ∈ 𝐴𝑖𝐶))
8076, 77, 79sylanbrc 695 . . . . . . . . . . . . . . 15 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶))
81 fnfvima 6400 . . . . . . . . . . . . . . 15 ((𝑆 Fn 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴 ∧ ⟨𝑖, 𝑘⟩ ∈ (𝐴𝐶)) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8269, 70, 80, 81syl3anc 1318 . . . . . . . . . . . . . 14 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆‘⟨𝑖, 𝑘⟩) ∈ (𝑆 “ (𝐴𝐶)))
8366, 82syl5eqel 2692 . . . . . . . . . . . . 13 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)))
84 elssuni 4403 . . . . . . . . . . . . 13 ((𝑖𝑆𝑘) ∈ (𝑆 “ (𝐴𝐶)) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
8583, 84syl 17 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝑆 “ (𝐴𝐶)))
867, 8, 59mrcssidd 16108 . . . . . . . . . . . . 13 (𝜑 (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8786ad2antrr 758 . . . . . . . . . . . 12 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑆 “ (𝐴𝐶)) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8885, 87sstrd 3578 . . . . . . . . . . 11 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → (𝑖𝑆𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
8965, 88eqsstrd 3602 . . . . . . . . . 10 (((𝜑𝑖𝐶) ∧ 𝑘 ∈ (𝐴 “ {𝑖})) → ((𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))‘𝑘) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9046, 50, 62, 89dprdlub 18248 . . . . . . . . 9 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
91 ovex 6577 . . . . . . . . . 10 (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ V
9291elpw 4114 . . . . . . . . 9 ((𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9390, 92sylibr 223 . . . . . . . 8 ((𝜑𝑖𝐶) → (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))) ∈ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9493, 29fmptd 6292 . . . . . . 7 (𝜑 → (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
95 frn 5966 . . . . . . 7 ((𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))):𝐶⟶𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
9694, 95syl 17 . . . . . 6 (𝜑 → ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))))
97 sspwuni 4547 . . . . . 6 (ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ 𝒫 (𝐾 (𝑆 “ (𝐴𝐶))) ↔ ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
9896, 97sylib 207 . . . . 5 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
997, 8mrcssvd 16106 . . . . 5 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (Base‘𝐺))
10098, 99sstrd 3578 . . . 4 (𝜑 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ⊆ (Base‘𝐺))
1017, 8, 43, 100mrcssd 16107 . . 3 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) ⊆ (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
1028mrcsscl 16103 . . . 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))))
1037, 98, 61, 102syl3anc 1318 . . 3 (𝜑 → (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) ⊆ (𝐾 (𝑆 “ (𝐴𝐶))))
104101, 103eqssd 3585 . 2 (𝜑 → (𝐾 (𝑆 “ (𝐴𝐶))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
105 eqid 2610 . . . . . . . 8 (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))
10691, 105dmmpti 5936 . . . . . . 7 dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼
107106a1i 11 . . . . . 6 (𝜑 → dom (𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) = 𝐼)
1081, 107, 44dprdres 18250 . . . . 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)))))))
109108simpld 474 . . . 4 (𝜑𝐺dom DProd ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶))
11044resmptd 5371 . . . 4 (𝜑 → ((𝑖𝐼 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))) ↾ 𝐶) = (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
111109, 110breqtrd 4609 . . 3 (𝜑𝐺dom DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗)))))
1128dprdspan 18249 . . 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))))))
113111, 112syl 17 . 2 (𝜑 → (𝐺 DProd (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))) = (𝐾 ran (𝑖𝐶 ↦ (𝐺 DProd (𝑗 ∈ (𝐴 “ {𝑖}) ↦ (𝑖𝑆𝑗))))))
114104, 113eqtr4d 2647 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Rel wrel 5043  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Grpcgrp 17245  SubGrpcsubg 17411   DProd cdprd 18215 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-cmn 18018  df-dprd 18217 This theorem is referenced by:  dprd2da  18264  dprd2db  18265
 Copyright terms: Public domain W3C validator