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Mirrors > Home > MPE Home > Th. List > tgcgrcomr | Structured version Visualization version GIF version |
Description: Congruence commutes on the RHS. 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 |
---|---|
tgcgrcomr | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrcomr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
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 | tgcgrcomr.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | tgcgrcomr.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
8 | 2, 3, 4, 5, 6, 7 | axtgcgrrflx 26256 | . 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
9 | 1, 8 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-trkgc 26242 df-trkg 26247 |
This theorem is referenced by: tgbtwnconn1lem1 26366 dfcgra2 26624 |
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