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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 26242 | . 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 26242 | . 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
12 | 1, 8, 11 | 3eqtr3d 2864 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-trkgc 26228 df-trkg 26233 |
This theorem is referenced by: tgcgrextend 26265 tgifscgr 26288 tgcgrsub 26289 iscgrglt 26294 trgcgrg 26295 tgcgrxfr 26298 cgr3swap12 26303 cgr3swap23 26304 tgbtwnxfr 26310 lnext 26347 tgbtwnconn1lem1 26352 tgbtwnconn1lem2 26353 tgbtwnconn1lem3 26354 tgbtwnconn1 26355 legov2 26366 legtri3 26370 legbtwn 26374 tgcgrsub2 26375 miriso 26450 mircgrextend 26462 mirtrcgr 26463 miduniq 26465 colmid 26468 symquadlem 26469 krippenlem 26470 midexlem 26472 ragcom 26478 ragflat 26484 ragcgr 26487 footexALT 26498 footexlem1 26499 footexlem2 26500 colperpexlem1 26510 mideulem2 26514 opphllem 26515 opphllem3 26529 lmiisolem 26576 hypcgrlem1 26579 trgcopy 26584 trgcopyeulem 26585 iscgra1 26590 cgracgr 26598 cgraswap 26600 cgrcgra 26601 cgracom 26602 cgratr 26603 flatcgra 26604 dfcgra2 26610 acopy 26613 acopyeu 26614 cgrg3col4 26633 tgsas1 26634 tgsas3 26637 tgasa1 26638 |
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