Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG) |
6 | tglineelsb2.2 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵) |
8 | tglineelsb2.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵) |
10 | tglineelsb2.4 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
11 | 1, 3, 2, 4, 8, 6, 10 | tglnssp 25247 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵) |
12 | 11 | sselda 3568 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵) |
13 | 10 | necomd 2837 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃) |
15 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄)) | |
16 | 1, 2, 3, 5, 7, 9, 12, 14, 15 | lncom 25317 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑄𝐿𝑃)) |
17 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝐺 ∈ TarskiG) |
18 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ∈ 𝐵) |
19 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑄 ∈ 𝐵) |
20 | 1, 3, 2, 4, 6, 8, 13 | tglnssp 25247 | . . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵) |
21 | 20 | sselda 3568 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵) |
22 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄) |
23 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃)) | |
24 | 1, 2, 3, 17, 18, 19, 21, 22, 23 | lncom 25317 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄)) |
25 | 16, 24 | impbida 873 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑄𝐿𝑃))) |
26 | 25 | eqrdv 2608 | 1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑄𝐿𝑃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 |
This theorem is referenced by: tglinethru 25331 coltr3 25343 footeq 25416 colperpexlem3 25424 mideulem2 25426 opphllem 25427 midex 25429 opphllem3 25441 opphllem5 25443 lmicom 25480 lmiisolem 25488 lnperpex 25495 trgcopy 25496 |
Copyright terms: Public domain | W3C validator |