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Theorem gsumpart 31897
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 2736 . . . 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 31566 . . 3 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1-onto𝐴)
121, 2, 3, 4, 5, 6, 11gsumf1o 19693 . 2 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))))
13 vsnex 5386 . . . . . . 7 {𝑥} ∈ V
1413a1i 11 . . . . . 6 ((𝜑𝑥𝑋) → {𝑥} ∈ V)
154adantr 481 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐴𝑉)
16 ssidd 3967 . . . . . . . . . 10 (𝜑𝐴𝐴)
1710, 16eqsstrd 3982 . . . . . . . . 9 (𝜑 𝑥𝑋 𝐶𝐴)
18 iunss 5005 . . . . . . . . 9 ( 𝑥𝑋 𝐶𝐴 ↔ ∀𝑥𝑋 𝐶𝐴)
1917, 18sylib 217 . . . . . . . 8 (𝜑 → ∀𝑥𝑋 𝐶𝐴)
2019r19.21bi 3234 . . . . . . 7 ((𝜑𝑥𝑋) → 𝐶𝐴)
2115, 20ssexd 5281 . . . . . 6 ((𝜑𝑥𝑋) → 𝐶 ∈ V)
2214, 21xpexd 7685 . . . . 5 ((𝜑𝑥𝑋) → ({𝑥} × 𝐶) ∈ V)
2322ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
24 iunexg 7896 . . . 4 ((𝑋𝑊 ∧ ∀𝑥𝑋 ({𝑥} × 𝐶) ∈ V) → 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
258, 23, 24syl2anc 584 . . 3 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ∈ V)
26 relxp 5651 . . . . . 6 Rel ({𝑥} × 𝐶)
2726a1i 11 . . . . 5 ((𝜑𝑥𝑋) → Rel ({𝑥} × 𝐶))
2827ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
29 reliun 5772 . . . 4 (Rel 𝑥𝑋 ({𝑥} × 𝐶) ↔ ∀𝑥𝑋 Rel ({𝑥} × 𝐶))
3028, 29sylibr 233 . . 3 (𝜑 → Rel 𝑥𝑋 ({𝑥} × 𝐶))
31 dmiun 5869 . . . . . 6 dom 𝑥𝑋 ({𝑥} × 𝐶) = 𝑥𝑋 dom ({𝑥} × 𝐶)
32 dmxpss 6123 . . . . . . . 8 dom ({𝑥} × 𝐶) ⊆ {𝑥}
3332rgenw 3068 . . . . . . 7 𝑥𝑋 dom ({𝑥} × 𝐶) ⊆ {𝑥}
34 ss2iun 4972 . . . . . . 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 3978 . . . . 5 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑥𝑋 {𝑥}
37 iunid 5020 . . . . 5 𝑥𝑋 {𝑥} = 𝑋
3836, 37sseqtri 3980 . . . 4 dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋
3938a1i 11 . . 3 (𝜑 → dom 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝑋)
40 fo2nd 7942 . . . . . . . 8 2nd :V–onto→V
41 fof 6756 . . . . . . . 8 (2nd :V–onto→V → 2nd :V⟶V)
4240, 41ax-mp 5 . . . . . . 7 2nd :V⟶V
43 ssv 3968 . . . . . . 7 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V
44 fssres 6708 . . . . . . 7 ((2nd :V⟶V ∧ 𝑥𝑋 ({𝑥} × 𝐶) ⊆ V) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V)
4542, 43, 44mp2an 690 . . . . . 6 (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶V
46 ffn 6668 . . . . . 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 31564 . . . . . . . 8 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝑥𝑋 𝐶)
49 imass2 6054 . . . . . . . 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 6029 . . . . . . . . . . 11 (2nd “ ∅) = ∅
52 xpeq1 5647 . . . . . . . . . . . . 13 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = (∅ × 𝑥𝑋 𝐶))
53 0xp 5730 . . . . . . . . . . . . 13 (∅ × 𝑥𝑋 𝐶) = ∅
5452, 53eqtrdi 2792 . . . . . . . . . . . 12 (𝑋 = ∅ → (𝑋 × 𝑥𝑋 𝐶) = ∅)
5554imaeq2d 6013 . . . . . . . . . . 11 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = (2nd “ ∅))
56 iuneq1 4970 . . . . . . . . . . . 12 (𝑋 = ∅ → 𝑥𝑋 𝐶 = 𝑥 ∈ ∅ 𝐶)
57 0iun 5023 . . . . . . . . . . . 12 𝑥 ∈ ∅ 𝐶 = ∅
5856, 57eqtrdi 2792 . . . . . . . . . . 11 (𝑋 = ∅ → 𝑥𝑋 𝐶 = ∅)
5951, 55, 583eqtr4a 2802 . . . . . . . . . 10 (𝑋 = ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6059adantl 482 . . . . . . . . 9 ((𝜑𝑋 = ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
61 2ndimaxp 31563 . . . . . . . . . 10 (𝑋 ≠ ∅ → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6261adantl 482 . . . . . . . . 9 ((𝜑𝑋 ≠ ∅) → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6360, 62pm2.61dane 3032 . . . . . . . 8 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝑥𝑋 𝐶)
6463, 10eqtrd 2776 . . . . . . 7 (𝜑 → (2nd “ (𝑋 × 𝑥𝑋 𝐶)) = 𝐴)
6550, 64sseqtrid 3996 . . . . . 6 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)) ⊆ 𝐴)
66 resssxp 6222 . . . . . 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 7049 . . . . 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 583 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
705, 69fcod 6694 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐵)
717, 4, 8, 9, 102ndresdju 31565 . . . 4 (𝜑 → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)–1-1𝐴)
722fvexi 6856 . . . . 5 0 ∈ V
7372a1i 11 . . . 4 (𝜑0 ∈ V)
745, 4fexd 7177 . . . 4 (𝜑𝐹 ∈ V)
756, 71, 73, 74fsuppco 9338 . . 3 (𝜑 → (𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶))) finSupp 0 )
761, 2, 3, 25, 30, 8, 39, 70, 75gsum2d 19749 . 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 3880 . . . . . . . . 9 𝑥𝑦 / 𝑥𝐶
78 csbeq1a 3869 . . . . . . . . 9 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
798, 21, 77, 78iunsnima2 31538 . . . . . . . 8 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) = 𝑦 / 𝑥𝐶)
80 df-ov 7360 . . . . . . . . 9 (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩)
8169ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (2nd 𝑥𝑋 ({𝑥} × 𝐶)): 𝑥𝑋 ({𝑥} × 𝐶)⟶𝐴)
82 simplr 767 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦𝑋)
83 vsnid 4623 . . . . . . . . . . . . . . 15 𝑦 ∈ {𝑦}
8483a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑦 ∈ {𝑦})
8579eleq2d 2823 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↔ 𝑧𝑦 / 𝑥𝐶))
8685biimpa 477 . . . . . . . . . . . . . 14 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → 𝑧𝑦 / 𝑥𝐶)
8784, 86opelxpd 5671 . . . . . . . . . . . . 13 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶))
88 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑥{𝑦}
8988, 77nfxp 5666 . . . . . . . . . . . . . . 15 𝑥({𝑦} × 𝑦 / 𝑥𝐶)
9089nfel2 2925 . . . . . . . . . . . . . 14 𝑥𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)
91 sneq 4596 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → {𝑥} = {𝑦})
9291, 78xpeq12d 5664 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ({𝑥} × 𝐶) = ({𝑦} × 𝑦 / 𝑥𝐶))
9392eleq2d 2823 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → (⟨𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶) ↔ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)))
9490, 93rspce 3570 . . . . . . . . . . . . 13 ((𝑦𝑋 ∧ ⟨𝑦, 𝑧⟩ ∈ ({𝑦} × 𝑦 / 𝑥𝐶)) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
9582, 87, 94syl2anc 584 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ∃𝑥𝑋𝑦, 𝑧⟩ ∈ ({𝑥} × 𝐶))
96 eliun 4958 . . . . . . . . . . . 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 6941 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)))
9997fvresd 6862 . . . . . . . . . . . 12 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = (2nd ‘⟨𝑦, 𝑧⟩))
100 vex 3449 . . . . . . . . . . . . 13 𝑦 ∈ V
101 vex 3449 . . . . . . . . . . . . 13 𝑧 ∈ V
102100, 101op2nd 7930 . . . . . . . . . . . 12 (2nd ‘⟨𝑦, 𝑧⟩) = 𝑧
10399, 102eqtrdi 2792 . . . . . . . . . . 11 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩) = 𝑧)
104103fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝐹‘((2nd 𝑥𝑋 ({𝑥} × 𝐶))‘⟨𝑦, 𝑧⟩)) = (𝐹𝑧))
10598, 104eqtrd 2776 . . . . . . . . 9 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → ((𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))‘⟨𝑦, 𝑧⟩) = (𝐹𝑧))
10680, 105eqtrid 2788 . . . . . . . 8 (((𝜑𝑦𝑋) ∧ 𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦})) → (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧) = (𝐹𝑧))
10779, 106mpteq12dva 5194 . . . . . . 7 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
1085adantr 481 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝐹:𝐴𝐵)
109 imassrn 6024 . . . . . . . . . 10 ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ ran 𝑥𝑋 ({𝑥} × 𝐶)
11010xpeq2d 5663 . . . . . . . . . . . . . 14 (𝜑 → (𝑋 × 𝑥𝑋 𝐶) = (𝑋 × 𝐴))
11148, 110sseqtrid 3996 . . . . . . . . . . . . 13 (𝜑 𝑥𝑋 ({𝑥} × 𝐶) ⊆ (𝑋 × 𝐴))
112 rnss 5894 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ ran (𝑋 × 𝐴))
115 rnxpss 6124 . . . . . . . . . . 11 ran (𝑋 × 𝐴) ⊆ 𝐴
116114, 115sstrdi 3956 . . . . . . . . . 10 ((𝜑𝑦𝑋) → ran 𝑥𝑋 ({𝑥} × 𝐶) ⊆ 𝐴)
117109, 116sstrid 3955 . . . . . . . . 9 ((𝜑𝑦𝑋) → ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ⊆ 𝐴)
11879, 117eqsstrrd 3983 . . . . . . . 8 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐶𝐴)
119108, 118feqresmpt 6911 . . . . . . 7 ((𝜑𝑦𝑋) → (𝐹𝑦 / 𝑥𝐶) = (𝑧𝑦 / 𝑥𝐶 ↦ (𝐹𝑧)))
120107, 119eqtr4d 2779 . . . . . 6 ((𝜑𝑦𝑋) → (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)) = (𝐹𝑦 / 𝑥𝐶))
121120oveq2d 7373 . . . . 5 ((𝜑𝑦𝑋) → (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧))) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
122121mpteq2dva 5205 . . . 4 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶))))
123 nfcv 2907 . . . . 5 𝑦(𝐺 Σg (𝐹𝐶))
124 nfcv 2907 . . . . . 6 𝑥𝐺
125 nfcv 2907 . . . . . 6 𝑥 Σg
126 nfcv 2907 . . . . . . 7 𝑥𝐹
127126, 77nfres 5939 . . . . . 6 𝑥(𝐹𝑦 / 𝑥𝐶)
128124, 125, 127nfov 7387 . . . . 5 𝑥(𝐺 Σg (𝐹𝑦 / 𝑥𝐶))
12978reseq2d 5937 . . . . . 6 (𝑥 = 𝑦 → (𝐹𝐶) = (𝐹𝑦 / 𝑥𝐶))
130129oveq2d 7373 . . . . 5 (𝑥 = 𝑦 → (𝐺 Σg (𝐹𝐶)) = (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
131123, 128, 130cbvmpt 5216 . . . 4 (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))) = (𝑦𝑋 ↦ (𝐺 Σg (𝐹𝑦 / 𝑥𝐶)))
132122, 131eqtr4di 2794 . . 3 (𝜑 → (𝑦𝑋 ↦ (𝐺 Σg (𝑧 ∈ ( 𝑥𝑋 ({𝑥} × 𝐶) “ {𝑦}) ↦ (𝑦(𝐹 ∘ (2nd 𝑥𝑋 ({𝑥} × 𝐶)))𝑧)))) = (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶))))
133132oveq2d 7373 . 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 2780 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥𝑋 ↦ (𝐺 Σg (𝐹𝐶)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  csb 3855  wss 3910  c0 4282  {csn 4586  cop 4592   ciun 4954  Disj wdisj 5070   class class class wbr 5105  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Rel wrel 5638   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  2nd c2nd 7920   finSupp cfsupp 9305  Basecbs 17083  0gc0g 17321   Σg cgsu 17322  CMndccmn 19562
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-disj 5071  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-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  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-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564
This theorem is referenced by:  elrspunidl  32203
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