![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tglineinteq | Structured version Visualization version GIF version |
Description: Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
Ref | Expression |
---|---|
tglineintmo.p | ⊢ 𝑃 = (Base‘𝐺) |
tglineintmo.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineintmo.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineintmo.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglineinteq.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tglineinteq.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tglineinteq.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tglineinteq.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tglineinteq.e | ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) |
tglineinteq.1 | ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐵)) |
tglineinteq.2 | ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐿𝐵)) |
tglineinteq.3 | ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝐷)) |
tglineinteq.4 | ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷)) |
Ref | Expression |
---|---|
tglineinteq | ⊢ (𝜑 → 𝑋 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineinteq.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐿𝐵)) | |
2 | tglineinteq.2 | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐴𝐿𝐵)) | |
3 | tglineintmo.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | tglineintmo.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | tglineintmo.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | tglineintmo.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
7 | tglineinteq.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | tglineinteq.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 3, 5, 4, 6, 7, 8, 1 | tglngne 25245 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | 3, 4, 5, 6, 7, 8, 9 | tgelrnln 25325 | . . 3 ⊢ (𝜑 → (𝐴𝐿𝐵) ∈ ran 𝐿) |
11 | tglineinteq.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
12 | tglineinteq.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
13 | tglineinteq.3 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶𝐿𝐷)) | |
14 | 3, 5, 4, 6, 11, 12, 13 | tglngne 25245 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
15 | 3, 4, 5, 6, 11, 12, 14 | tgelrnln 25325 | . . 3 ⊢ (𝜑 → (𝐶𝐿𝐷) ∈ ran 𝐿) |
16 | tglineinteq.e | . . . 4 ⊢ (𝜑 → ¬ (𝐴 ∈ (𝐵𝐿𝐶) ∨ 𝐵 = 𝐶)) | |
17 | 3, 4, 5, 6, 7, 8, 11, 12, 16 | tglineneq 25339 | . . 3 ⊢ (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷)) |
18 | 3, 4, 5, 6, 10, 15, 17 | tglineintmo 25337 | . 2 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷))) |
19 | 1, 13 | jca 553 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷))) |
20 | tglineinteq.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷)) | |
21 | 2, 20 | jca 553 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷))) |
22 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑋 ∈ (𝐴𝐿𝐵))) | |
23 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑋 ∈ (𝐶𝐿𝐷))) | |
24 | 22, 23 | anbi12d 743 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷)))) |
25 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑌 ∈ (𝐴𝐿𝐵))) | |
26 | eleq1 2676 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑌 ∈ (𝐶𝐿𝐷))) | |
27 | 25, 26 | anbi12d 743 | . . 3 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷)))) |
28 | 24, 27 | moi 3356 | . 2 ⊢ (((𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐴𝐿𝐵)) ∧ ∃*𝑥(𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ∧ ((𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷)) ∧ (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷)))) → 𝑋 = 𝑌) |
29 | 1, 2, 18, 19, 21, 28 | syl212anc 1328 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 ‘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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
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-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-s2 13444 df-s3 13445 df-trkgc 25147 df-trkgb 25148 df-trkgcb 25149 df-trkg 25152 df-cgrg 25206 |
This theorem is referenced by: symquadlem 25384 midexlem 25387 outpasch 25447 hlpasch 25448 tgasa1 25539 |
Copyright terms: Public domain | W3C validator |