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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 | . . 3 ⊢ (𝜑 → 𝑋 ≠ 𝐴) | |
2 | mirhl2.2 | . . 3 ⊢ (𝜑 → 𝑌 ≠ 𝐴) | |
3 | mirval.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
4 | mirval.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | mirval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | mirval.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
7 | mirval.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | mirval.s | . . . . 5 ⊢ 𝑆 = (pInvG‘𝐺) | |
9 | mirhl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
10 | mirhl.m | . . . . 5 ⊢ 𝑀 = (𝑆‘𝐴) | |
11 | mirhl.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
12 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mircl 26029 | . . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃) |
13 | mirhl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 5 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → 𝐺 ∈ TarskiG) |
15 | 9 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → 𝐴 ∈ 𝑃) |
16 | 11 | adantr 474 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → 𝑌 ∈ 𝑃) |
17 | simpr 479 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → (𝑀‘𝑌) = 𝐴) | |
18 | 3, 6, 4, 7, 8, 14, 15, 10 | mircinv 26036 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → (𝑀‘𝐴) = 𝐴) |
19 | 17, 18 | eqtr4d 2817 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → (𝑀‘𝑌) = (𝑀‘𝐴)) |
20 | 3, 6, 4, 7, 8, 14, 15, 10, 16, 15, 19 | mireq 26033 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑀‘𝑌) = 𝐴) → 𝑌 = 𝐴) |
21 | 20 | ex 403 | . . . . . 6 ⊢ (𝜑 → ((𝑀‘𝑌) = 𝐴 → 𝑌 = 𝐴)) |
22 | 21 | necon3d 2990 | . . . . 5 ⊢ (𝜑 → (𝑌 ≠ 𝐴 → (𝑀‘𝑌) ≠ 𝐴)) |
23 | 2, 22 | mpd 15 | . . . 4 ⊢ (𝜑 → (𝑀‘𝑌) ≠ 𝐴) |
24 | mirhl2.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) | |
25 | 3, 6, 4, 5, 13, 9, 12, 24 | tgbtwncom 25856 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑋)) |
26 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mirbtwn 26026 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌)) |
27 | 3, 4, 5, 12, 9, 13, 11, 23, 25, 26 | tgbtwnconn2 25944 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))) |
28 | 1, 2, 27 | 3jca 1119 | . 2 ⊢ (𝜑 → (𝑋 ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋)))) |
29 | mirhl.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
30 | 3, 4, 29, 13, 11, 9, 5 | ishlg 25970 | . 2 ⊢ (𝜑 → (𝑋(𝐾‘𝐴)𝑌 ↔ (𝑋 ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))))) |
31 | 28, 30 | mpbird 249 | 1 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 Basecbs 16266 distcds 16358 TarskiGcstrkg 25798 Itvcitv 25804 LineGclng 25805 hlGchlg 25968 pInvGcmir 26020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-xnn0 11720 df-z 11734 df-uz 11998 df-fz 12649 df-fzo 12790 df-hash 13442 df-word 13606 df-concat 13667 df-s1 13692 df-s2 14005 df-s3 14006 df-trkgc 25816 df-trkgb 25817 df-trkgcb 25818 df-trkg 25821 df-cgrg 25879 df-hlg 25969 df-mir 26021 |
This theorem is referenced by: colhp 26135 sacgr 26196 sacgrOLD 26197 |
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