lmiopp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lmiopp		Structured version Visualization version GIF version

Theorem lmiopp 27067

Description: Line mirroring produces points on the opposite side of the mirroring line. Theorem 10.14 of [Schwabhauser] p. 92. (Contributed by Thierry Arnoux, 2-Aug-2020.)

**Hypotheses**
Ref	Expression
lmiopp.p	⊢ 𝑃 = (Base‘𝐺)
lmiopp.m	⊢ − = (dist‘𝐺)
lmiopp.i	⊢ 𝐼 = (Itv‘𝐺)
lmiopp.l	⊢ 𝐿 = (LineG‘𝐺)
lmiopp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
lmiopp.h	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmiopp.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lmiopp.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
lmiopp.n	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmiopp.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
lmiopp.1	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)

**Assertion**
Ref	Expression
lmiopp	⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Distinct variable groups: − ,𝑎,𝑏,𝑡 𝐴,𝑎,𝑏,𝑡 𝐷,𝑎,𝑏,𝑡 𝐺,𝑎,𝑏,𝑡 𝐼,𝑎,𝑏,𝑡 𝑀,𝑎,𝑏,𝑡 𝑡,𝑂 𝑃,𝑎,𝑏,𝑡 𝜑,𝑎,𝑏,𝑡 Allowed substitution hints: 𝐿(𝑡,𝑎,𝑏) 𝑂(𝑎,𝑏)

**Proof of Theorem lmiopp**
Step	Hyp	Ref	Expression
1		lmiopp.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		lmiopp.m	. 2 ⊢ − = (dist‘𝐺)
3		lmiopp.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		lmiopp.o	. 2 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		lmiopp.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		lmiopp.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		lmiopp.h	. . 3 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
8		lmiopp.n	. . 3 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
9		lmiopp.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
10		lmiopp.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
11	1, 2, 3, 6, 7, 8, 9, 10, 5	lmicl 27051	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		eqidd 2739	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
13	1, 2, 3, 6, 7, 8, 9, 10, 5, 11	islmib 27052	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴)))))
14	12, 13	mpbid 231	. . 3 ⊢ (𝜑 → ((𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿(𝑀‘𝐴)) ∨ 𝐴 = (𝑀‘𝐴))))
15	14	simpld 494	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ 𝐷)
16		lmiopp.1	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
17	1, 2, 3, 6, 7, 8, 9, 10, 5	lmilmi 27054	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
18	17	eqeq1d 2740	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
19	1, 2, 3, 6, 7, 8, 9, 10, 11	lmiinv 27057	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
20		eqcom 2745	. . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
22	18, 19, 21	3bitr3d 308	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
23	1, 2, 3, 6, 7, 8, 9, 10, 5	lmiinv 27057	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
24	22, 23	bitrd 278	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
25	16, 24	mtbird 324	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
26	1, 2, 3, 6, 7, 5, 11	midbtwn 27044	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27	1, 2, 3, 4, 5, 11, 15, 16, 25, 26	islnoppd 27005	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 ∖ cdif 3880 class class class wbr 5070 {copab 5132 ran crn 5581 ‘cfv 6418 (class class class)co 7255 2c2 11958 Basecbs 16840 distcds 16897 TarskiGcstrkg 26693 Dim_TarskiG≥cstrkgld 26697 Itvcitv 26699 LineGclng 26700 ⟂Gcperpg 26960 midGcmid 27037 lInvGclmi 27038

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-concat 14202 df-s1 14229 df-s2 14489 df-s3 14490 df-trkgc 26713 df-trkgb 26714 df-trkgcb 26715 df-trkgld 26717 df-trkg 26718 df-cgrg 26776 df-leg 26848 df-mir 26918 df-rag 26959 df-perpg 26961 df-mid 27039 df-lmi 27040

This theorem is referenced by: trgcopyeulem 27070

W3C validator