![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 7267 | . . 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 7270 | . . 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 7271 | . . 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 205 = wceq 1542 Isom wiso 6502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 |
This theorem is referenced by: resisoeq45d 41766 |
Copyright terms: Public domain | W3C validator |