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Mirrors > Home > MPE Home > Th. List > hlcomd | Structured version Visualization version GIF version |
Description: The half-line relation commutes. Theorem 6.6 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
ishlg.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
hlcomd.1 | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) |
Ref | Expression |
---|---|
hlcomd | ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcomd.1 | . 2 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐵) | |
2 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
5 | ishlg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | ishlg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | ishlg.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | ishlg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
9 | 2, 3, 4, 5, 6, 7, 8 | hlcomb 26317 | . 2 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐵 ↔ 𝐵(𝐾‘𝐶)𝐴)) |
10 | 1, 9 | mpbid 233 | 1 ⊢ (𝜑 → 𝐵(𝐾‘𝐶)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5058 ‘cfv 6349 Basecbs 16473 Itvcitv 26150 hlGchlg 26314 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
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 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-hlg 26315 |
This theorem is referenced by: hlcgreulem 26331 opphllem4 26464 opphllem5 26465 opphl 26468 hlpasch 26470 lnopp2hpgb 26477 colhp 26484 cgrahl1 26530 cgrahl2 26531 cgrahl 26541 cgracol 26542 dfcgra2 26544 sacgr 26545 acopy 26547 acopyeu 26548 inaghl 26559 tgasa1 26572 |
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