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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 | . 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 28684 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ 𝑃) |
13 | mirhl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃) | |
14 | 3, 6, 4, 7, 8, 5, 9, 10, 11, 2 | mirne 28690 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ≠ 𝐴) |
15 | mirhl2.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝑋𝐼(𝑀‘𝑌))) | |
16 | 3, 6, 4, 5, 13, 9, 12, 15 | tgbtwncom 28511 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑋)) |
17 | 3, 6, 4, 7, 8, 5, 9, 10, 11 | mirbtwn 28681 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝑌)𝐼𝑌)) |
18 | 3, 4, 5, 12, 9, 13, 11, 14, 16, 17 | tgbtwnconn2 28599 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))) |
19 | mirhl.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
20 | 3, 4, 19, 13, 11, 9, 5 | ishlg 28625 | . 2 ⊢ (𝜑 → (𝑋(𝐾‘𝐴)𝑌 ↔ (𝑋 ≠ 𝐴 ∧ 𝑌 ≠ 𝐴 ∧ (𝑋 ∈ (𝐴𝐼𝑌) ∨ 𝑌 ∈ (𝐴𝐼𝑋))))) |
21 | 1, 2, 18, 20 | mpbir3and 1341 | 1 ⊢ (𝜑 → 𝑋(𝐾‘𝐴)𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 hlGchlg 28623 pInvGcmir 28675 |
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-rmo 3378 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 df-hlg 28624 df-mir 28676 |
This theorem is referenced by: colhp 28793 sacgr 28854 |
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