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Mirrors > Home > MPE Home > Th. List > tgcgrneq | 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)) |
tgcgrneq.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
tgcgrneq | ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcgrneq.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | tkgeom.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tkgeom.d | . . . 4 ⊢ − = (dist‘𝐺) | |
4 | tkgeom.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | tgcgrcomlr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | tgcgrcomlr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
8 | tgcgrcomlr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | tgcgrcomlr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
10 | tgcgrcomlr.6 | . . . 4 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | tgcgreqb 25792 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
12 | 11 | necon3bid 3042 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
13 | 1, 12 | mpbid 224 | 1 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 distcds 16313 TarskiGcstrkg 25741 Itvcitv 25747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-nul 5012 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-iota 6085 df-fv 6130 df-ov 6907 df-trkgc 25759 df-trkg 25764 |
This theorem is referenced by: hlcgrex 25927 midexlem 26003 footex 26029 mideulem2 26042 opphllem3 26057 trgcopy 26112 iscgra1 26118 cgrane1 26120 cgrane2 26121 cgrcgra 26129 cgrg3col4 26151 tgsas2 26154 tgsas3 26155 tgasa1 26156 tgsss1 26158 |
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