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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 28338 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴)) |
8 | tgcgrcomr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
9 | 7, 8 | eqtr3d 2767 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 distcds 17245 TarskiGcstrkg 28303 Itvcitv 28309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-trkgc 28324 df-trkg 28329 |
This theorem is referenced by: hlcgrex 28492 dfcgra2 28706 |
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