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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 462 | . . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
4 | 3 | rexbidv 3174 | . . 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 27531 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
14 | 5, 6, 7, 8, 9, 10, 12, 11 | hpgbr 27531 | . . 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 231 | 1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ∖ cdif 3906 class class class wbr 5104 {copab 5166 ran crn 5633 ‘cfv 6494 (class class class)co 7352 Basecbs 17043 TarskiGcstrkg 27198 Itvcitv 27204 LineGclng 27205 hpGchpg 27528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-hpg 27529 |
This theorem is referenced by: trgcopyeulem 27576 tgasa1 27629 |
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