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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 7031 | . . . 4 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁)) | |
3 | 2 | oveq1d 7038 | . . 3 ⊢ (𝑛 = 𝑁 → ((0...𝑛) ↑𝑚 (0...𝑀)) = ((0...𝑁) ↑𝑚 (0...𝑀))) |
4 | eqeq2 2808 | . . 3 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ (0...𝑀)(𝑐‘𝑗) = 𝑁)) | |
5 | 3, 4 | rabeqbidv 3433 | . 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 7055 | . . . 4 ⊢ ((0...𝑁) ↑𝑚 (0...𝑀)) ∈ V | |
8 | 7 | rabex 5133 | . . 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 6664 | 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 1525 ∈ wcel 2083 {crab 3111 Vcvv 3440 ↦ cmpt 5047 ‘cfv 6232 (class class class)co 7023 ↑𝑚 cmap 8263 0cc0 10390 ℕ0cn0 11751 ...cfz 12746 Σcsu 14880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-iota 6196 df-fun 6234 df-fv 6240 df-ov 7026 |
This theorem is referenced by: etransclem16 42099 etransclem24 42107 etransclem26 42109 etransclem28 42111 etransclem31 42114 etransclem32 42115 etransclem34 42117 etransclem35 42118 etransclem37 42120 etransclem38 42121 |
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