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| Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem5 | Structured version Visualization version GIF version | ||
| Description: A change of bound variable, often used in proofs for etransc 46298. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
| Ref | Expression |
|---|---|
| etransclem5 | ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝑗) = (𝑦 − 𝑗)) | |
| 2 | 1 | oveq1d 7446 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) |
| 3 | 2 | cbvmptv 5255 | . . 3 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) |
| 4 | oveq2 7439 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝑦 − 𝑗) = (𝑦 − 𝑘)) | |
| 5 | eqeq1 2741 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 = 0 ↔ 𝑘 = 0)) | |
| 6 | 5 | ifbid 4549 | . . . . 5 ⊢ (𝑗 = 𝑘 → if(𝑗 = 0, (𝑃 − 1), 𝑃) = if(𝑘 = 0, (𝑃 − 1), 𝑃)) |
| 7 | 4, 6 | oveq12d 7449 | . . . 4 ⊢ (𝑗 = 𝑘 → ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)) = ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃))) |
| 8 | 7 | mpteq2dv 5244 | . . 3 ⊢ (𝑗 = 𝑘 → (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) |
| 9 | 3, 8 | eqtrid 2789 | . 2 ⊢ (𝑗 = 𝑘 → (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃))) = (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) |
| 10 | 9 | cbvmptv 5255 | 1 ⊢ (𝑗 ∈ (0...𝑀) ↦ (𝑥 ∈ 𝑋 ↦ ((𝑥 − 𝑗)↑if(𝑗 = 0, (𝑃 − 1), 𝑃)))) = (𝑘 ∈ (0...𝑀) ↦ (𝑦 ∈ 𝑋 ↦ ((𝑦 − 𝑘)↑if(𝑘 = 0, (𝑃 − 1), 𝑃)))) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ifcif 4525 ↦ cmpt 5225 (class class class)co 7431 0cc0 11155 1c1 11156 − cmin 11492 ...cfz 13547 ↑cexp 14102 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: etransclem27 46276 etransclem29 46278 etransclem31 46280 etransclem32 46281 etransclem33 46282 etransclem34 46283 etransclem35 46284 etransclem38 46287 etransclem40 46289 etransclem42 46291 etransclem44 46293 etransclem45 46294 etransclem46 46295 |
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