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Mirrors > Home > MPE Home > Th. List > tgcgreqb | 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)) |
Ref | Expression |
---|---|
tgcgreqb | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . 3 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
6 | tgcgrcomlr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝑃) |
8 | tgcgrcomlr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃) |
10 | tgcgrcomlr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
12 | tgcgrcomlr.6 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
14 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
15 | 14 | oveq1d 7160 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐵 − 𝐵)) |
16 | 13, 15 | eqtr3d 2855 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 − 𝐷) = (𝐵 − 𝐵)) |
17 | 1, 2, 3, 5, 7, 9, 11, 16 | axtgcgrid 26176 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷) |
18 | 4 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐺 ∈ TarskiG) |
19 | tgcgrcomlr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
20 | 19 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 ∈ 𝑃) |
21 | 10 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐵 ∈ 𝑃) |
22 | 8 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ∈ 𝑃) |
23 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
24 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) | |
25 | 24 | oveq1d 7160 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 − 𝐷) = (𝐷 − 𝐷)) |
26 | 23, 25 | eqtrd 2853 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐷)) |
27 | 1, 2, 3, 18, 20, 21, 22, 26 | axtgcgrid 26176 | . 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵) |
28 | 17, 27 | impbida 797 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-trkgc 26161 df-trkg 26166 |
This theorem is referenced by: tgcgreq 26195 tgcgrneq 26196 |
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