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Mirrors > Home > MPE Home > Th. List > Mathboxes > isoeq145d | Structured version Visualization version GIF version |
Description: Equality deduction for isometries. (Contributed by RP, 14-Jan-2025.) |
Ref | Expression |
---|---|
isoeq145.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
isoeq145.4 | ⊢ (𝜑 → 𝐴 = 𝐶) |
isoeq145.5 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
isoeq145d | ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isoeq145.1 | . . 3 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | isoeq1 7335 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) |
4 | isoeq145.4 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
5 | isoeq4 7338 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵))) |
7 | isoeq145.5 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
8 | isoeq5 7339 | . . 3 ⊢ (𝐵 = 𝐷 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))) |
10 | 3, 6, 9 | 3bitrd 305 | 1 ⊢ (𝜑 → (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 Isom wiso 6560 |
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 2707 |
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 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 |
This theorem is referenced by: resisoeq45d 43411 |
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