![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25552 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴)) |
8 | tgcgrcomr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
9 | 7, 8 | eqtr3d 2788 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1624 ∈ wcel 2131 ‘cfv 6041 (class class class)co 6805 Basecbs 16051 distcds 16144 TarskiGcstrkg 25520 Itvcitv 25526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-nul 4933 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ral 3047 df-rex 3048 df-rab 3051 df-v 3334 df-sbc 3569 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-iota 6004 df-fv 6049 df-ov 6808 df-trkgc 25538 df-trkg 25543 |
This theorem is referenced by: hlcgrex 25702 dfcgra2 25912 |
Copyright terms: Public domain | W3C validator |