Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tglineelsb2 | Structured version Visualization version GIF version |
Description: If 𝑆 lies on PQ , then PQ = PS . Theorem 6.16 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) |
tglineelsb2.3 | ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
tglineelsb2.5 | ⊢ (𝜑 → 𝑆 ≠ 𝑃) |
tglineelsb2.6 | ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄)) |
Ref | Expression |
---|---|
tglineelsb2 | ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑃𝐿𝑆)) |
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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝐺 ∈ TarskiG) |
6 | tglineelsb2.1 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ∈ 𝐵) |
8 | tglineelsb2.3 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐵) | |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑆 ∈ 𝐵) |
10 | tglineelsb2.5 | . . . . . 6 ⊢ (𝜑 → 𝑆 ≠ 𝑃) | |
11 | 10 | necomd 3073 | . . . . 5 ⊢ (𝜑 → 𝑃 ≠ 𝑆) |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑃 ≠ 𝑆) |
13 | tglineelsb2.2 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ 𝐵) |
15 | tglineelsb2.4 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
16 | 15 | necomd 3073 | . . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃) |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ≠ 𝑃) |
18 | tglineelsb2.6 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (𝑃𝐿𝑄)) | |
19 | 18 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑆 ∈ (𝑃𝐿𝑄)) |
20 | 1, 2, 3, 5, 14, 7, 9, 17, 19 | lncom 26410 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑆 ∈ (𝑄𝐿𝑃)) |
21 | 1, 2, 3, 5, 7, 9, 14, 12, 20, 17 | lnrot1 26411 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑄 ∈ (𝑃𝐿𝑆)) |
22 | 1, 3, 2, 4, 6, 13, 15 | tglnssp 26340 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑄) ⊆ 𝐵) |
23 | 22 | sselda 3969 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ 𝐵) |
24 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑄)) | |
25 | 1, 2, 3, 5, 7, 9, 12, 14, 17, 21, 23, 24 | tglineeltr 26419 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑄)) → 𝑥 ∈ (𝑃𝐿𝑆)) |
26 | 4 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝐺 ∈ TarskiG) |
27 | 6 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑃 ∈ 𝐵) |
28 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑄 ∈ 𝐵) |
29 | 15 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑃 ≠ 𝑄) |
30 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑆 ∈ 𝐵) |
31 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑆 ≠ 𝑃) |
32 | 18 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑆 ∈ (𝑃𝐿𝑄)) |
33 | 1, 3, 2, 4, 6, 8, 11 | tglnssp 26340 | . . . . 5 ⊢ (𝜑 → (𝑃𝐿𝑆) ⊆ 𝐵) |
34 | 33 | sselda 3969 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑥 ∈ 𝐵) |
35 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑥 ∈ (𝑃𝐿𝑆)) | |
36 | 1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 34, 35 | tglineeltr 26419 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑃𝐿𝑆)) → 𝑥 ∈ (𝑃𝐿𝑄)) |
37 | 25, 36 | impbida 799 | . 2 ⊢ (𝜑 → (𝑥 ∈ (𝑃𝐿𝑄) ↔ 𝑥 ∈ (𝑃𝐿𝑆))) |
38 | 37 | eqrdv 2821 | 1 ⊢ (𝜑 → (𝑃𝐿𝑄) = (𝑃𝐿𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13925 df-s1 13952 df-s2 14212 df-s3 14213 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 df-cgrg 26299 |
This theorem is referenced by: tglinethru 26424 ncolncol 26434 coltr3 26436 hlperpnel 26513 colperpexlem3 26520 mideulem2 26522 lmieu 26572 lmiisolem 26584 |
Copyright terms: Public domain | W3C validator |