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Mirrors > Home > MPE Home > Th. List > hpgcom | Structured version Visualization version GIF version |
Description: The half-plane relation commutes. Theorem 9.12 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
hpgid.p | ⊢ 𝑃 = (Base‘𝐺) |
hpgid.i | ⊢ 𝐼 = (Itv‘𝐺) |
hpgid.l | ⊢ 𝐿 = (LineG‘𝐺) |
hpgid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hpgid.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
hpgid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
hpgid.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
hpgcom.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
hpgcom.1 | ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) |
Ref | Expression |
---|---|
hpgcom | ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hpgcom.1 | . 2 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) | |
2 | ancom 460 | . . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
4 | 3 | rexbidv 3177 | . . 3 ⊢ (𝜑 → (∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
5 | hpgid.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
6 | hpgid.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
7 | hpgid.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | hpgid.o | . . . 4 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
9 | hpgid.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
10 | hpgid.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
11 | hpgid.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | hpgcom.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
13 | 5, 6, 7, 8, 9, 10, 11, 12 | hpgbr 28783 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
14 | 5, 6, 7, 8, 9, 10, 12, 11 | hpgbr 28783 | . . 3 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
15 | 4, 13, 14 | 3bitr4d 311 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐵((hpG‘𝐺)‘𝐷)𝐴)) |
16 | 1, 15 | mpbid 232 | 1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∖ cdif 3960 class class class wbr 5148 {copab 5210 ran crn 5690 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 hpGchpg 28780 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-hpg 28781 |
This theorem is referenced by: trgcopyeulem 28828 tgasa1 28881 |
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