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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 26270 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
12 | 1, 11 | mpbid 234 | 1 ⊢ (𝜑 → 𝐶 = 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 distcds 16577 TarskiGcstrkg 26219 Itvcitv 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-trkgc 26237 df-trkg 26242 |
This theorem is referenced by: tgcgrextend 26274 tgidinside 26360 tgbtwnconn1lem3 26363 krippenlem 26479 ragcgr 26496 lmiisolem 26585 cgrg3col4 26642 |
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