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| Mirrors > Home > MPE Home > Th. List > tgcgrcomlr | Structured version Visualization version GIF version | ||
| Description: Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-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 |
|---|---|
| tgcgrcomlr | ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tgcgrcomlr.6 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷)) | |
| 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 | 2, 3, 4, 5, 6, 7 | axtgcgrrflx 28470 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴)) |
| 9 | tgcgrcomlr.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 10 | tgcgrcomlr.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
| 11 | 2, 3, 4, 5, 9, 10 | axtgcgrrflx 28470 | . 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
| 12 | 1, 8, 11 | 3eqtr3d 2785 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-trkgc 28456 df-trkg 28461 |
| This theorem is referenced by: tgcgrextend 28493 tgifscgr 28516 tgcgrsub 28517 iscgrglt 28522 trgcgrg 28523 tgcgrxfr 28526 cgr3swap12 28531 cgr3swap23 28532 tgbtwnxfr 28538 lnext 28575 tgbtwnconn1lem1 28580 tgbtwnconn1lem2 28581 tgbtwnconn1lem3 28582 tgbtwnconn1 28583 legov2 28594 legtri3 28598 legbtwn 28602 tgcgrsub2 28603 miriso 28678 mircgrextend 28690 mirtrcgr 28691 miduniq 28693 colmid 28696 symquadlem 28697 krippenlem 28698 midexlem 28700 ragcom 28706 ragflat 28712 ragcgr 28715 footexALT 28726 footexlem1 28727 footexlem2 28728 colperpexlem1 28738 mideulem2 28742 opphllem 28743 opphllem3 28757 lmiisolem 28804 hypcgrlem1 28807 trgcopy 28812 trgcopyeulem 28813 iscgra1 28818 cgracgr 28826 cgraswap 28828 cgrcgra 28829 cgracom 28830 cgratr 28831 flatcgra 28832 dfcgra2 28838 acopy 28841 acopyeu 28842 cgrg3col4 28861 tgsas1 28862 tgsas3 28865 tgasa1 28866 |
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