|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > tgcgreq | Structured version Visualization version GIF version | ||
| Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) | 
| Ref | Expression | 
|---|---|
| tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) | 
| tkgeom.d | ⊢ − = (dist‘𝐺) | 
| tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| tgcgrcomlr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| tgcgrcomlr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| tgcgrcomlr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) | 
| tgcgrcomlr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) | 
| tgcgrcomlr.6 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | 
| tgcgreq.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Ref | Expression | 
|---|---|
| tgcgreq | ⊢ (𝜑 → 𝐶 = 𝐷) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tgcgreq.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
| 4 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 5 | tkgeom.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 6 | tgcgrcomlr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | tgcgrcomlr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 8 | tgcgrcomlr.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 9 | tgcgrcomlr.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
| 10 | tgcgrcomlr.6 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
| 11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | tgcgreqb 28489 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) | 
| 12 | 1, 11 | mpbid 232 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 | 
| 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 ax-nul 5306 | 
| 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-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 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-iota 6514 df-fv 6569 df-ov 7434 df-trkgc 28456 df-trkg 28461 | 
| This theorem is referenced by: tgcgrextend 28493 tgidinside 28579 tgbtwnconn1lem3 28582 krippenlem 28698 ragcgr 28715 lmiisolem 28804 cgrg3col4 28861 | 
| Copyright terms: Public domain | W3C validator |