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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 7439 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
| 3 | 2 | oveq1d 7446 | . . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑m (0...𝑀)) = ((0...𝑁) ↑m (0...𝑀))) |
| 4 | eqeq2 2749 | . . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)) | |
| 5 | 3, 4 | rabeqbidv 3455 | . 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 7464 | . . . 4 ⊢ ((0...𝑁) ↑m (0...𝑀)) ∈ V | |
| 8 | 7 | rabex 5339 | . . 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 7039 | 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 ℕ0cn0 12526 ...cfz 13547 Σcsu 15722 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: etransclem16 46265 etransclem24 46273 etransclem26 46275 etransclem28 46277 etransclem31 46280 etransclem32 46281 etransclem34 46283 etransclem35 46284 etransclem37 46286 etransclem38 46287 |
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