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Theorem gsumpart 31946
Description: Express a group sum as a double sum, grouping along a (possibly infinite) partition. (Contributed by Thierry Arnoux, 22-Jun-2024.)
Hypotheses
Ref Expression
gsumpart.b 𝐵 = (Base‘𝐺)
gsumpart.z 0 = (0g𝐺)
gsumpart.g (𝜑𝐺 ∈ CMnd)
gsumpart.a (𝜑𝐴𝑉)
gsumpart.x (𝜑𝑋𝑊)
gsumpart.f (𝜑𝐹:𝐴𝐵)
gsumpart.w (𝜑𝐹 finSupp 0 )
gsumpart.1 (𝜑Disj 𝑥𝑋 𝐶)
gsumpart.2 (𝜑 𝑥𝑋 𝐶 = 𝐴)
Assertion
Ref Expression
gsumpart (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)   0 (𝑥)

Proof of Theorem gsumpart
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumpart.b . . 3 𝐵 = (Base‘𝐺)
2 gsumpart.z . . 3 0 = (0g𝐺)
3 gsumpart.g . . 3 (𝜑𝐺 ∈ CMnd)
4 gsumpart.a . . 3 (𝜑𝐴𝑉)
5 gsumpart.f . . 3 (𝜑𝐹:𝐴𝐵)
6 gsumpart.w . . 3 (𝜑𝐹 finSupp 0 )
7 eqid 2733 . . . 4 𝑥𝑋 ({𝑥} × 𝐶) = 𝑥𝑋 ({𝑥} × 𝐶)
8 gsumpart.x . . . 4 (𝜑𝑋𝑊)
9 gsumpart.1 . . . 4 (𝜑Disj 𝑥𝑋 𝐶)
10 gsumpart.2 . . . 4 (𝜑 𝑥𝑋 𝐶 = 𝐴)
117, 4, 8, 9, 102ndresdjuf1o 31612 . . 3 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1-onto𝐴)
121, 2, 3, 4, 5, 6, 11gsumf1o 19698 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))))
13 vsnex 5387 . . . . . . 7 {𝑥} ∈ V
1413a1i 11 . . . . . 6 ((𝜑𝑥𝑋) → {𝑥} ∈ V)
154adantr 482 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴𝑉)
16 ssidd 3968 . . . . . . . . . 10 (𝜑𝐴𝐴)
1710, 16eqsstrd 3983 . . . . . . . . 9 (𝜑 𝑥𝑋 𝐶𝐴)
18 iunss 5006 . . . . . . . . 9 ( 𝑥𝑋 𝐶𝐴 ↔ ∀𝑥𝑋 𝐶𝐴)
1917, 18sylib 217 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 𝐶𝐴)
2019r19.21bi 3233 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐶𝐴)
2115, 20ssexd 5282 . . . . . 6 ((𝜑𝑥𝑋) → 𝐶 ∈ V)
2214, 21xpexd 7686 . . . . 5 ((𝜑𝑥𝑋) → ({𝑥} × 𝐶) ∈ V)
2322ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
24 iunexg 7897 . . . 4 ((𝑋𝑊 ∧ ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V) → 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
258, 23, 24syl2anc 585 . . 3 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
26 relxp 5652 . . . . . 6 Rel ({𝑥} × 𝐶)
2726a1i 11 . . . . 5 ((𝜑𝑥𝑋) → Rel ({𝑥} × 𝐶))
2827ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
29 reliun 5773 . . . 4 (Rel 𝑥𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
3028, 29sylibr 233 . . 3 (𝜑 → Rel 𝑥𝑋 ({𝑥} × 𝐶))
31 dmiun 5870 . . . . . 6 dom 𝑥𝑋 ({𝑥} × 𝐶) = 𝑥𝑋 dom ({𝑥} × 𝐶)
32 dmxpss 6124 . . . . . . . 8 dom ({𝑥} × 𝐶) ⊆ {𝑥}
3332rgenw 3065 . . . . . . 7 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34 ss2iun 4973 . . . . . . 7 (∀𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥} → 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥})
3533, 34ax-mp 5 . . . . . 6 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥}
3631, 35eqsstri 3979 . . . . 5 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥}
37 iunid 5021 . . . . 5 𝑥𝑋 {𝑥} = 𝑋
3836, 37sseqtri 3981 . . . 4 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
3938a1i 11 . . 3 (𝜑 → dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40 fo2nd 7943 . . . . . . . 8 2nd :V–onto→V
41 fof 6757 . . . . . . . 8 (2nd :V–onto→V → 2nd :V⟶V)
4240, 41ax-mp 5 . . . . . . 7 2nd :V⟶V
43 ssv 3969 . . . . . . 7 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V
44 fssres 6709 . . . . . . 7 ((2nd :V⟶V ∧ 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V)
4542, 43, 44mp2an 691 . . . . . 6 (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V
46 ffn 6669 . . . . . 6 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶))
4745, 46mp1i 13 . . . . 5 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶))
48 djussxp2 31610 . . . . . . . 8 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝑥𝑋 𝐶)
49 imass2 6055 . . . . . . . 8 ( 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝑥𝑋 𝐶) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × 𝑥𝑋 𝐶)))
5048, 49ax-mp 5 . . . . . . 7 (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ (2nd “ (𝑋 × 𝑥𝑋 𝐶))
51 ima0 6030 . . . . . . . . . . 11 (2nd “ ∅) = ∅
52 xpeq1 5648 . . . . . . . . . . . . 13 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = (∅ × 𝑥𝑋 𝐶))
53 0xp 5731 . . . . . . . . . . . . 13 (∅ × 𝑥𝑋 𝐶) = ∅
5452, 53eqtrdi 2789 . . . . . . . . . . . 12 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = ∅)
5554imaeq2d 6014 . . . . . . . . . . 11 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = (2nd “ ∅))
56 iuneq1 4971 . . . . . . . . . . . 12 (𝑋 = ∅ → 𝑥𝑋 𝐶 = 𝑥 ∈ ∅ 𝐶)
57 0iun 5024 . . . . . . . . . . . 12 𝑥 ∈ ∅ 𝐶 = ∅
5856, 57eqtrdi 2789 . . . . . . . . . . 11 (𝑋 = ∅ → 𝑥𝑋 𝐶 = ∅)
5951, 55, 583eqtr4a 2799 . . . . . . . . . 10 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6059adantl 483 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
61 2ndimaxp 31609 . . . . . . . . . 10 (𝑋 ≠ ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6261adantl 483 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6360, 62pm2.61dane 3029 . . . . . . . 8 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6463, 10eqtrd 2773 . . . . . . 7 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝐴)
6550, 64sseqtrid 3997 . . . . . 6 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66 resssxp 6223 . . . . . 6 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴 ↔ (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴))
6765, 66sylib 217 . . . . 5 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴))
68 dff2 7050 . . . . 5 ((2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴 ↔ ((2nd 𝑥𝑋 ({𝑥} × 𝐶)) Fn 𝑥𝑋 ({𝑥} × 𝐶) ∧ (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ ( 𝑥𝑋 ({𝑥} × 𝐶) × 𝐴)))
6947, 67, 68sylanbrc 584 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
705, 69fcod 6695 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐵)
717, 4, 8, 9, 102ndresdju 31611 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1𝐴)
722fvexi 6857 . . . . 5 0 ∈ V
7372a1i 11 . . . 4 (𝜑0 ∈ V)
745, 4fexd 7178 . . . 4 (𝜑𝐹 ∈ V)
756, 71, 73, 74fsuppco 9343 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))) finSupp 0 )
761, 2, 3, 25, 30, 8, 39, 70, 75gsum2d 19754 . 2 (𝜑 → (𝐺 Σg (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))) = (𝐺 Σg (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))))))
77 nfcsb1v 3881 . . . . . . . . 9 𝑥𝑦 / 𝑥𝐶
78 csbeq1a 3870 . . . . . . . . 9 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
798, 21, 77, 78iunsnima2 31584 . . . . . . . 8 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) = 𝑦 / 𝑥𝐶)
80 df-ov 7361 . . . . . . . . 9 (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
8169ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
82 simplr 768 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦𝑋)
83 vsnid 4624 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑦}
8483a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
8579eleq2d 2820 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↔ 𝑧𝑦 / 𝑥𝐶))
8685biimpa 478 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑧𝑦 / 𝑥𝐶)
8784, 86opelxpd 5672 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶))
88 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑥{𝑦}
8988, 77nfxp 5667 . . . . . . . . . . . . . . 15 𝑥({𝑦} × 𝑦 / 𝑥𝐶)
9089nfel2 2922 . . . . . . . . . . . . . 14 𝑥𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)
91 sneq 4597 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
9291, 78xpeq12d 5665 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × 𝑦 / 𝑥𝐶))
9392eleq2d 2820 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)))
9490, 93rspce 3569 . . . . . . . . . . . . 13 ((𝑦𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
9582, 87, 94syl2anc 585 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96 eliun 4959 . . . . . . . . . . . 12 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝑋 ({𝑥} × 𝐶) ↔ ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
9795, 96sylibr 233 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ 𝑥𝑋 ({𝑥} × 𝐶))
9881, 97fvco3d 6942 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)))
9997fvresd 6863 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100 vex 3448 . . . . . . . . . . . . 13 𝑦 ∈ V
101 vex 3448 . . . . . . . . . . . . 13 𝑧 ∈ V
102100, 101op2nd 7931 . . . . . . . . . . . 12 (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
10399, 102eqtrdi 2789 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104103fveq2d 6847 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹𝑧))
10598, 104eqtrd 2773 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹𝑧))
10680, 105eqtrid 2785 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹𝑧))
10779, 106mpteq12dva 5195 . . . . . . 7 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
1085adantr 482 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝐹:𝐴𝐵)
109 imassrn 6025 . . . . . . . . . 10 ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran 𝑥𝑋 ({𝑥} × 𝐶)
11010xpeq2d 5664 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 × 𝑥𝑋 𝐶) = (𝑋 × 𝐴))
11148, 110sseqtrid 3997 . . . . . . . . . . . . 13 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112 rnss 5895 . . . . . . . . . . . . 13 ( 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
113111, 112syl 17 . . . . . . . . . . . 12 (𝜑 → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
114113adantr 482 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115 rnxpss 6125 . . . . . . . . . . 11 ran (𝑋 × 𝐴) ⊆ 𝐴
116114, 115sstrdi 3957 . . . . . . . . . 10 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117109, 116sstrid 3956 . . . . . . . . 9 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
11879, 117eqsstrrd 3984 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐶𝐴)
119108, 118feqresmpt 6912 . . . . . . 7 ((𝜑𝑦𝑋) → (𝐹𝑦 / 𝑥𝐶) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
120107, 119eqtr4d 2776 . . . . . 6 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝐹𝑦 / 𝑥𝐶))
121120oveq2d 7374 . . . . 5 ((𝜑𝑦𝑋) → (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
122121mpteq2dva 5206 . . . 4 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶))))
123 nfcv 2904 . . . . 5 𝑦(𝐺 Σg (𝐹𝐶))
124 nfcv 2904 . . . . . 6 𝑥𝐺
125 nfcv 2904 . . . . . 6 𝑥 Σg
126 nfcv 2904 . . . . . . 7 𝑥𝐹
127126, 77nfres 5940 . . . . . 6 𝑥(𝐹𝑦 / 𝑥𝐶)
128124, 125, 127nfov 7388 . . . . 5 𝑥(𝐺 Σg (𝐹𝑦 / 𝑥𝐶))
12978reseq2d 5938 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝐶) = (𝐹𝑦 / 𝑥𝐶))
130129oveq2d 7374 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
131123, 128, 130cbvmpt 5217 . . . 4 (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
132122, 131eqtr4di 2791 . . 3 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))))
133132oveq2d 7374 . 2 (𝜑 → (𝐺 Σg (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))))) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
13412, 76, 1333eqtrd 2777 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  Vcvv 3444  csb 3856  wss 3911  c0 4283  {csn 4587  cop 4593   ciun 4955  Disj wdisj 5071   class class class wbr 5106  cmpt 5189   × cxp 5632  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  ccom 5638  Rel wrel 5639   Fn wfn 6492  wf 6493  ontowfo 6495  cfv 6497  (class class class)co 7358  2nd c2nd 7921   finSupp cfsupp 9308  Basecbs 17088  0gc0g 17326   Σg cgsu 17327  CMndccmn 19567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-disj 5072  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-seq 13913  df-hash 14237  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-gsum 17329  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569
This theorem is referenced by:  elrspunidl  32251
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