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| Mirrors > Home > MPE Home > Th. List > tgcgrcoml | Structured version Visualization version GIF version | ||
| Description: Congruence commutes on the LHS. Variant of Theorem 2.5 of [Schwabhauser] p. 27, but in a convenient form for a common case. (Contributed by David A. Wheeler, 29-Jun-2020.) |
| Ref | Expression |
|---|---|
| tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
| tkgeom.d | ⊢ − = (dist‘𝐺) |
| tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tgcgrcomr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| tgcgrcomr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| tgcgrcomr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
| tgcgrcomr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
| tgcgrcomr.6 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
| Ref | Expression |
|---|---|
| tgcgrcoml | ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
| 3 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | tkgeom.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 5 | tgcgrcomr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 6 | tgcgrcomr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 7 | 1, 2, 3, 4, 5, 6 | axtgcgrrflx 28470 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴)) |
| 8 | tgcgrcomr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
| 9 | 7, 8 | eqtr3d 2779 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (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: hlcgrex 28624 dfcgra2 28838 |
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