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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 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | 1 | fveq2d 6677 | . . . 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 483 | . . . . 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 26338 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
13 | 12 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
14 | mirln.m | . . . . 5 ⊢ 𝑀 = (𝑆‘𝐴) | |
15 | 3, 4, 5, 6, 7, 9, 13, 14 | mircinv 26457 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴) |
16 | 2, 15 | eqtr3d 2861 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴) |
17 | 11 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) |
18 | 16, 17 | eqeltrd 2916 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
19 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG) |
20 | 12 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃) |
21 | mirln.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
22 | 3, 6, 5, 8, 10, 21 | tglnpt 26338 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
23 | 22 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃) |
24 | 3, 4, 5, 6, 7, 19, 20, 14, 23 | mircl 26450 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃) |
25 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
26 | 3, 4, 5, 6, 7, 8, 12, 14, 22 | mirbtwn 26447 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
27 | 26 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
28 | 3, 5, 6, 19, 20, 23, 24, 25, 27 | btwnlng2 26409 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ (𝐴𝐿𝐵)) |
29 | 10 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿) |
30 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷) |
31 | 21 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷) |
32 | 3, 5, 6, 19, 20, 23, 25, 25, 29, 30, 31 | tglinethru 26425 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐴𝐿𝐵)) |
33 | 28, 32 | eleqtrrd 2919 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
34 | 18, 33 | pm2.61dane 3107 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ran crn 5559 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 distcds 16577 TarskiGcstrkg 26219 Itvcitv 26225 LineGclng 26226 pInvGcmir 26441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-dju 9333 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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 13926 df-s1 13953 df-s2 14213 df-s3 14214 df-trkgc 26237 df-trkgb 26238 df-trkgcb 26239 df-trkg 26242 df-cgrg 26300 df-mir 26442 |
This theorem is referenced by: opphllem2 26537 opphllem4 26539 colhp 26559 |
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