Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tgcgrcomimp | Structured version Visualization version GIF version |
Description: Congruence commutes on the RHS. Theorem 2.5 of [Schwabhauser] p. 27. (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) |
tgcgrcomimp.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgcgrcomimp.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgcgrcomimp.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgcgrcomimp.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
Ref | Expression |
---|---|
tgcgrcomimp | ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tgcgrcomimp.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
6 | tgcgrcomimp.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
7 | 1, 2, 3, 4, 5, 6 | axtgcgrrflx 26248 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
8 | 7 | eqeq2d 2832 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
9 | 8 | biimpd 231 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-trkgc 26234 df-trkg 26239 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |