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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 26256 | . 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 26256 | . 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶)) |
12 | 1, 8, 11 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-trkgc 26242 df-trkg 26247 |
This theorem is referenced by: tgcgrextend 26279 tgifscgr 26302 tgcgrsub 26303 iscgrglt 26308 trgcgrg 26309 tgcgrxfr 26312 cgr3swap12 26317 cgr3swap23 26318 tgbtwnxfr 26324 lnext 26361 tgbtwnconn1lem1 26366 tgbtwnconn1lem2 26367 tgbtwnconn1lem3 26368 tgbtwnconn1 26369 legov2 26380 legtri3 26384 legbtwn 26388 tgcgrsub2 26389 miriso 26464 mircgrextend 26476 mirtrcgr 26477 miduniq 26479 colmid 26482 symquadlem 26483 krippenlem 26484 midexlem 26486 ragcom 26492 ragflat 26498 ragcgr 26501 footexALT 26512 footexlem1 26513 footexlem2 26514 colperpexlem1 26524 mideulem2 26528 opphllem 26529 opphllem3 26543 lmiisolem 26590 hypcgrlem1 26593 trgcopy 26598 trgcopyeulem 26599 iscgra1 26604 cgracgr 26612 cgraswap 26614 cgrcgra 26615 cgracom 26616 cgratr 26617 flatcgra 26618 dfcgra2 26624 acopy 26627 acopyeu 26628 cgrg3col4 26647 tgsas1 26648 tgsas3 26651 tgasa1 26652 |
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