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Mirrors > Home > MPE Home > Th. List > tgcgreq | Structured version Visualization version GIF version |
Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgcgrcomlr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgcgrcomlr.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgcgrcomlr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgcgrcomlr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgcgrcomlr.6 | ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
tgcgreq.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
tgcgreq | ⊢ (𝜑 → 𝐶 = 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgreq.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
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 | tgcgrcomlr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | tgcgrcomlr.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | tgcgrcomlr.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | tgcgrcomlr.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | tgcgrcomlr.6 | . . 3 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | tgcgreqb 27252 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
12 | 1, 11 | mpbid 231 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 distcds 17102 TarskiGcstrkg 27198 Itvcitv 27204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-nul 5262 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 df-ov 7355 df-trkgc 27219 df-trkg 27224 |
This theorem is referenced by: tgcgrextend 27256 tgidinside 27342 tgbtwnconn1lem3 27345 krippenlem 27461 ragcgr 27478 lmiisolem 27567 cgrg3col4 27624 |
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