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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 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 28560 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵) |
| 12 | 11 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵) |
| 13 | 10 | necomd 2996 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃) |
| 14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃) |
| 15 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄)) | |
| 16 | 1, 2, 3, 5, 7, 9, 12, 14, 15 | lncom 28630 | . . 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 28560 | . . . . 5 ⊢ (𝜑 → (𝑄𝐿𝑃) ⊆ 𝐵) |
| 21 | 20 | sselda 3983 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ 𝐵) |
| 22 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑃 ≠ 𝑄) |
| 23 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑄𝐿𝑃)) | |
| 24 | 1, 2, 3, 17, 18, 19, 21, 22, 23 | lncom 28630 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑄𝐿𝑃)) → 𝑥 ∈ (𝑃𝐿𝑄)) |
| 25 | 16, 24 | impbida 801 | . 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 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: tglinethru 28644 coltr3 28656 footeq 28732 colperpexlem3 28740 mideulem2 28742 opphllem 28743 midex 28745 opphllem3 28757 opphllem5 28759 lmicom 28796 lmiisolem 28804 lnperpex 28811 trgcopy 28812 |
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