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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 25781 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴)) |
8 | tgcgrcomr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
9 | 7, 8 | eqtr3d 2863 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 distcds 16321 TarskiGcstrkg 25749 Itvcitv 25755 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-iota 6090 df-fv 6135 df-ov 6913 df-trkgc 25767 df-trkg 25772 |
This theorem is referenced by: hlcgrex 25935 dfcgra2 26145 |
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