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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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG) |
6 | tglineelsb2.2 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵) |
8 | tglineelsb2.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵) |
10 | tglineelsb2.4 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
11 | 1, 3, 2, 4, 8, 6, 10 | tglnssp 26332 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵) |
12 | 11 | sselda 3966 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵) |
13 | 10 | necomd 3071 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃) |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃) |
15 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄)) | |
16 | 1, 2, 3, 5, 7, 9, 12, 14, 15 | lncom 26402 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃)) |
17 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG) |
18 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ∈ 𝐵) |
19 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄 ∈ 𝐵) |
20 | 1, 3, 2, 4, 6, 8, 13 | tglnssp 26332 | . . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵) |
21 | 20 | sselda 3966 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵) |
22 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄) |
23 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃)) | |
24 | 1, 2, 3, 17, 18, 19, 21, 22, 23 | lncom 26402 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄)) |
25 | 16, 24 | impbida 799 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃))) |
26 | 25 | eqrdv 2819 | 1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 TarskiGcstrkg 26210 Itvcitv 26216 LineGclng 26217 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: tglinethru 26416 coltr3 26428 footeq 26504 colperpexlem3 26512 mideulem2 26514 opphllem 26515 midex 26517 opphllem3 26529 opphllem5 26531 lmicom 26568 lmiisolem 26576 lnperpex 26583 trgcopy 26584 |
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