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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 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
6 | tgcgrcomlr.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝑃) |
8 | tgcgrcomlr.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃) |
10 | tgcgrcomlr.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
12 | tgcgrcomlr.6 | . . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
13 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
14 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
15 | 14 | oveq1d 7173 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐵 − 𝐵)) |
16 | 13, 15 | eqtr3d 2860 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 − 𝐷) = (𝐵 − 𝐵)) |
17 | 1, 2, 3, 5, 7, 9, 11, 16 | axtgcgrid 26251 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷) |
18 | 4 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐺 ∈ TarskiG) |
19 | tgcgrcomlr.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
20 | 19 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 ∈ 𝑃) |
21 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐵 ∈ 𝑃) |
22 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ∈ 𝑃) |
23 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐶 − 𝐷)) |
24 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷) | |
25 | 24 | oveq1d 7173 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 − 𝐷) = (𝐷 − 𝐷)) |
26 | 23, 25 | eqtrd 2858 | . . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐷)) |
27 | 1, 2, 3, 18, 20, 21, 22, 26 | axtgcgrid 26251 | . 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵) |
28 | 17, 27 | impbida 799 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-trkgc 26236 df-trkg 26241 |
This theorem is referenced by: tgcgreq 26270 tgcgrneq 26271 |
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