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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 28573 | . . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
10 | 3, 4, 5, 6, 7, 8, 9 | tgelrnln 28653 | . . 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 28573 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷) |
15 | 3, 4, 5, 6, 11, 12, 14 | tgelrnln 28653 | . . 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 28667 | . . 3 ⊢ (𝜑 → (𝐴𝐿𝐵) ≠ (𝐶𝐿𝐷)) |
18 | 3, 4, 5, 6, 10, 15, 17 | tglineintmo 28665 | . 2 ⊢ (𝜑 → ∃*𝑥(𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷))) |
19 | 1, 13 | jca 511 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷))) |
20 | tglineinteq.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐶𝐿𝐷)) | |
21 | 2, 20 | jca 511 | . 2 ⊢ (𝜑 → (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷))) |
22 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑋 ∈ (𝐴𝐿𝐵))) | |
23 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑋 ∈ (𝐶𝐿𝐷))) | |
24 | 22, 23 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑋 ∈ (𝐴𝐿𝐵) ∧ 𝑋 ∈ (𝐶𝐿𝐷)))) |
25 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐴𝐿𝐵) ↔ 𝑌 ∈ (𝐴𝐿𝐵))) | |
26 | eleq1 2827 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝑥 ∈ (𝐶𝐿𝐷) ↔ 𝑌 ∈ (𝐶𝐿𝐷))) | |
27 | 25, 26 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝑌 → ((𝑥 ∈ (𝐴𝐿𝐵) ∧ 𝑥 ∈ (𝐶𝐿𝐷)) ↔ (𝑌 ∈ (𝐴𝐿𝐵) ∧ 𝑌 ∈ (𝐶𝐿𝐷)))) |
28 | 24, 27 | moi 3727 | . 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 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∃*wmo 2536 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 df-cgrg 28534 |
This theorem is referenced by: symquadlem 28712 midexlem 28715 outpasch 28778 hlpasch 28779 tgasa1 28881 |
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