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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 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
2 | 1 | fveq2d 6889 | . . . 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 480 | . . . . 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 28308 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
13 | 12 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
14 | mirln.m | . . . . 5 ⊢ 𝑀 = (𝑆‘𝐴) | |
15 | 3, 4, 5, 6, 7, 9, 13, 14 | mircinv 28427 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐴) = 𝐴) |
16 | 2, 15 | eqtr3d 2768 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) = 𝐴) |
17 | 11 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) |
18 | 16, 17 | eqeltrd 2827 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
19 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG) |
20 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃) |
21 | mirln.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
22 | 3, 6, 5, 8, 10, 21 | tglnpt 28308 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
23 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃) |
24 | 3, 4, 5, 6, 7, 19, 20, 14, 23 | mircl 28420 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝑃) |
25 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ≠ 𝐵) | |
26 | 3, 4, 5, 6, 7, 8, 12, 14, 22 | mirbtwn 28417 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
27 | 26 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)) |
28 | 3, 5, 6, 19, 20, 23, 24, 25, 27 | btwnlng2 28379 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ (𝐴𝐿𝐵)) |
29 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿) |
30 | 11 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷) |
31 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷) |
32 | 3, 5, 6, 19, 20, 23, 25, 25, 29, 30, 31 | tglinethru 28395 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐴𝐿𝐵)) |
33 | 28, 32 | eleqtrrd 2830 | . 2 ⊢ ((𝜑 ∧ 𝐴 ≠ 𝐵) → (𝑀‘𝐵) ∈ 𝐷) |
34 | 18, 33 | pm2.61dane 3023 | 1 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ran crn 5670 ‘cfv 6537 (class class class)co 7405 Basecbs 17153 distcds 17215 TarskiGcstrkg 28186 Itvcitv 28192 LineGclng 28193 pInvGcmir 28411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-oadd 8471 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-s2 14805 df-s3 14806 df-trkgc 28207 df-trkgb 28208 df-trkgcb 28209 df-trkg 28212 df-cgrg 28270 df-mir 28412 |
This theorem is referenced by: opphllem2 28507 opphllem4 28509 colhp 28529 |
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