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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 26408 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
8 | 7 | eqeq2d 2749 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
9 | 8 | biimpd 232 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 distcds 16677 TarskiGcstrkg 26376 Itvcitv 26382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-iota 6297 df-fv 6347 df-ov 7173 df-trkgc 26394 df-trkg 26399 |
This theorem is referenced by: (None) |
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