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Mirrors > Home > MPE Home > Th. List > mirln | Structured version Visualization version GIF version |
Description: If two points are on the same line, so is the mirror point of one through the other. (Contributed by Thierry Arnoux, 21-Dec-2019.) |
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) |
mirln.m | ⊢ 𝑀 = (𝑆‘𝐴) |
mirln.1 | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
mirln.a | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
mirln.b | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
Ref | Expression |
---|---|
mirln | ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | 1 | fveq2d 6668 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = (𝑀‘𝐵)) |
3 | mirval.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
4 | mirval.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
5 | mirval.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | mirval.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
7 | mirval.s | . . . . 5 ⊢ 𝑆 = (pInvG‘𝐺) | |
8 | mirval.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
10 | mirln.1 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
11 | mirln.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
12 | 3, 6, 5, 8, 10, 11 | tglnpt 26263 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
13 | 12 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
14 | mirln.m | . . . . 5 ⊢ 𝑀 = (𝑆‘𝐴) | |
15 | 3, 4, 5, 6, 7, 9, 13, 14 | mircinv 26382 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴) |
16 | 2, 15 | eqtr3d 2858 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴) |
17 | 11 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) |
18 | 16, 17 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
19 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG) |
20 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃) |
21 | mirln.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
22 | 3, 6, 5, 8, 10, 21 | tglnpt 26263 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
23 | 22 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃) |
24 | 3, 4, 5, 6, 7, 19, 20, 14, 23 | mircl 26375 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃) |
25 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
26 | 3, 4, 5, 6, 7, 8, 12, 14, 22 | mirbtwn 26372 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
27 | 26 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
28 | 3, 5, 6, 19, 20, 23, 24, 25, 27 | btwnlng2 26334 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ (𝐴𝐿𝐵)) |
29 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿) |
30 | 11 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷) |
31 | 21 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷) |
32 | 3, 5, 6, 19, 20, 23, 25, 25, 29, 30, 31 | tglinethru 26350 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐴𝐿𝐵)) |
33 | 28, 32 | eleqtrrd 2916 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
34 | 18, 33 | pm2.61dane 3104 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3016 ran crn 5550 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 distcds 16564 TarskiGcstrkg 26144 Itvcitv 26150 LineGclng 26151 pInvGcmir 26366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-xnn0 11957 df-z 11971 df-uz 12233 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-concat 13913 df-s1 13940 df-s2 14200 df-s3 14201 df-trkgc 26162 df-trkgb 26163 df-trkgcb 26164 df-trkg 26167 df-cgrg 26225 df-mir 26367 |
This theorem is referenced by: opphllem2 26462 opphllem4 26464 colhp 26484 |
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