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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 28137 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
8 | 7 | eqeq2d 2735 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) ↔ (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
9 | 8 | biimpd 228 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐶 − 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6533 (class class class)co 7401 Basecbs 17140 distcds 17202 TarskiGcstrkg 28102 Itvcitv 28108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-nul 5296 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 df-trkgc 28123 df-trkg 28128 |
This theorem is referenced by: (None) |
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