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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem12 | Structured version Visualization version GIF version |
Description: 𝐶 applied to 𝑁. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem12.1 | ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) |
etransclem12.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
etransclem12 | ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem12.1 | . 2 ⊢ 𝐶 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛}) | |
2 | oveq2 7423 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
3 | 2 | oveq1d 7430 | . . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀))) |
4 | eqeq2 2737 | . . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)) | |
5 | 3, 4 | rabeqbidv 3437 | . 2 ⊢ (𝑛 = 𝑁 → {𝑐 ∈ ((0...𝑛) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛} = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) |
6 | etransclem12.2 | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
7 | ovex 7448 | . . . 4 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ V | |
8 | 7 | rabex 5329 | . . 3 ⊢ {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ V |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁} ∈ V) |
10 | 1, 5, 6, 9 | fvmptd3 7022 | 1 ⊢ (𝜑 → (𝐶‘𝑁) = {𝑐 ∈ ((0...𝑁) ↑m (0...𝑀)) ∣ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ↦ cmpt 5226 ‘cfv 6542 (class class class)co 7415 ↑m cmap 8841 0cc0 11136 ℕ0cn0 12500 ...cfz 13514 Σcsu 15662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7418 |
This theorem is referenced by: etransclem16 45700 etransclem24 45708 etransclem26 45710 etransclem28 45712 etransclem31 45715 etransclem32 45716 etransclem34 45718 etransclem35 45719 etransclem37 45721 etransclem38 45722 |
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