Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mirhl2 | Structured version Visualization version GIF version |
Description: Deduce half-line relation from mirror point. (Contributed by Thierry Arnoux, 8-Aug-2020.) |
Ref | Expression |
---|---|
mirval.p | ⊢ 𝑃 = (Base‘𝐺) |
mirval.d | ⊢ − = (dist‘𝐺) |
mirval.i | ⊢ 𝐼 = (Itv‘𝐺) |
mirval.l | ⊢ 𝐿 = (LineG‘𝐺) |
mirval.s | ⊢ 𝑆 = (pInvG‘𝐺) |
mirval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
mirhl.m | ⊢ 𝑀 = (𝑆‘𝐴) |
mirhl.k | ⊢ 𝐾 = (hlG‘𝐺) |
mirhl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
mirhl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
mirhl.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
mirhl.z | ⊢ (𝜑 → 𝑍 ∈ 𝑃) |
mirhl2.1 | ⊢ (𝜑 → 𝑋 ≠ 𝐴) |
mirhl2.2 | ⊢ (𝜑 → 𝑌 ≠ 𝐴) |
mirhl2.3 | ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) |
Ref | Expression |
---|---|
mirhl2 | ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mirhl2.1 | . 2 ⊢ (𝜑 → 𝑋 ≠ 𝐴) | |
2 | mirhl2.2 | . 2 ⊢ (𝜑 → 𝑌 ≠ 𝐴) | |
3 | mirval.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
4 | mirval.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | mirval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | mirval.d | . . . 4 ⊢ − = (dist‘𝐺) | |
7 | mirval.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . . 4 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirhl.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | mirhl.m | . . . 4 ⊢ 𝑀 = (𝑆‘𝐴) | |
11 | mirhl.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
12 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mircl 26449 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃) |
13 | mirhl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 3, 6, 4, 7, 8, 5, 9, 10, 11, 2 | mirne 26455 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ≠ 𝐴) |
15 | mirhl2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) | |
16 | 3, 6, 4, 5, 13, 9, 12, 15 | tgbtwncom 26276 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑋)) |
17 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mirbtwn 26446 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌)) |
18 | 3, 4, 5, 12, 9, 13, 11, 14, 16, 17 | tgbtwnconn2 26364 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))) |
19 | mirhl.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
20 | 3, 4, 19, 13, 11, 9, 5 | ishlg 26390 | . 2 ⊢ (𝜑 → (𝑋(𝐾‘𝐴)𝑌 ↔ (𝑋 ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))))) |
21 | 1, 2, 18, 20 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 hlGchlg 26388 pInvGcmir 26440 |
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 df-hlg 26389 df-mir 26441 |
This theorem is referenced by: colhp 26558 sacgr 26619 |
Copyright terms: Public domain | W3C validator |