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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 466 | . . . . 5 ⊢ ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐)) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) ↔ (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
4 | 3 | rexbidv 3052 | . . 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 25652 | . . 3 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) |
14 | 5, 6, 7, 8, 9, 10, 12, 11 | hpgbr 25652 | . . 3 ⊢ (𝜑 → (𝐵((hpG‘𝐺)‘𝐷)𝐴 ↔ ∃𝑐 ∈ 𝑃 (𝐵𝑂𝑐 ∧ 𝐴𝑂𝑐))) |
15 | 4, 13, 14 | 3bitr4d 300 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ 𝐵((hpG‘𝐺)‘𝐷)𝐴)) |
16 | 1, 15 | mpbid 222 | 1 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∖ cdif 3571 class class class wbr 4653 {copab 4712 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 hpGchpg 25649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-hpg 25650 |
This theorem is referenced by: trgcopyeulem 25697 tgasa1 25739 |
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