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Mirrors > Home > MPE Home > Th. List > tglinecom | Structured version Visualization version GIF version |
Description: Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglineelsb2.1 | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
tglineelsb2.2 | ⊢ (𝜑 → 𝑄 ∈ 𝐵) |
tglineelsb2.4 | ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
Ref | Expression |
---|---|
tglinecom | ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tglineelsb2.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | tglineelsb2.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG) |
6 | tglineelsb2.2 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵) |
8 | tglineelsb2.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵) |
10 | tglineelsb2.4 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
11 | 1, 3, 2, 4, 8, 6, 10 | tglnssp 27322 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵) |
12 | 11 | sselda 3942 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵) |
13 | 10 | necomd 2997 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃) |
15 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄)) | |
16 | 1, 2, 3, 5, 7, 9, 12, 14, 15 | lncom 27392 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃)) |
17 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG) |
18 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ∈ 𝐵) |
19 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄 ∈ 𝐵) |
20 | 1, 3, 2, 4, 6, 8, 13 | tglnssp 27322 | . . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵) |
21 | 20 | sselda 3942 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵) |
22 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄) |
23 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃)) | |
24 | 1, 2, 3, 17, 18, 19, 21, 22, 23 | lncom 27392 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄)) |
25 | 16, 24 | impbida 799 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃))) |
26 | 25 | eqrdv 2735 | 1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 TarskiGcstrkg 27197 Itvcitv 27203 LineGclng 27204 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-trkgc 27218 df-trkgb 27219 df-trkgcb 27220 df-trkg 27223 |
This theorem is referenced by: tglinethru 27406 coltr3 27418 footeq 27494 colperpexlem3 27502 mideulem2 27504 opphllem 27505 midex 27507 opphllem3 27519 opphllem5 27521 lmicom 27558 lmiisolem 27566 lnperpex 27573 trgcopy 27574 |
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