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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 26907 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | 3, 4, 5, 6, 7, 8, 9 | tgelrnln 26987 | . . 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 26907 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
15 | 3, 4, 5, 6, 11, 12, 14 | tgelrnln 26987 | . . 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 27001 | . . 3 ⊢ (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷)) |
18 | 3, 4, 5, 6, 10, 15, 17 | tglineintmo 26999 | . 2 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷))) |
19 | 1, 13 | jca 512 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷))) |
20 | tglineinteq.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷)) | |
21 | 2, 20 | jca 512 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷))) |
22 | eleq1 2828 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑋 ∈ (𝐴𝐿𝐵))) | |
23 | eleq1 2828 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑋 ∈ (𝐶𝐿𝐷))) | |
24 | 22, 23 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷)))) |
25 | eleq1 2828 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑌 ∈ (𝐴𝐿𝐵))) | |
26 | eleq1 2828 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑌 ∈ (𝐶𝐿𝐷))) | |
27 | 25, 26 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷)))) |
28 | 24, 27 | moi 3657 | . 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 1379 | 1 ⊢ (𝜑 → 𝑋 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1542 ∈ wcel 2110 ∃*wmo 2540 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 TarskiGcstrkg 26784 Itvcitv 26790 LineGclng 26791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-oadd 8290 df-er 8479 df-pm 8599 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-dju 9658 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-n0 12232 df-xnn0 12304 df-z 12318 df-uz 12580 df-fz 13237 df-fzo 13380 df-hash 14041 df-word 14214 df-concat 14270 df-s1 14297 df-s2 14557 df-s3 14558 df-trkgc 26805 df-trkgb 26806 df-trkgcb 26807 df-trkg 26810 df-cgrg 26868 |
This theorem is referenced by: symquadlem 27046 midexlem 27049 outpasch 27112 hlpasch 27113 tgasa1 27215 |
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