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Theorem lmiopp 28828

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 28812	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
12		eqidd 2741	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) = (𝑀‘𝐴))
13	1, 2, 3, 6, 7, 8, 9, 10, 5, 11	islmib 28813	. . . 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 232	. . 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 28815	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐴)) = 𝐴)
18	17	eqeq1d 2742	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ 𝐴 = (𝑀‘𝐴)))
19	1, 2, 3, 6, 7, 8, 9, 10, 11	lmiinv 28818	. . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑀‘𝐴)) = (𝑀‘𝐴) ↔ (𝑀‘𝐴) ∈ 𝐷))
20		eqcom 2747	. . . . . 6 ⊢ (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴)
21	20	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐴) ↔ (𝑀‘𝐴) = 𝐴))
22	18, 19, 21	3bitr3d 309	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ (𝑀‘𝐴) = 𝐴))
23	1, 2, 3, 6, 7, 8, 9, 10, 5	lmiinv 28818	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) = 𝐴 ↔ 𝐴 ∈ 𝐷))
24	22, 23	bitrd 279	. . 3 ⊢ (𝜑 → ((𝑀‘𝐴) ∈ 𝐷 ↔ 𝐴 ∈ 𝐷))
25	16, 24	mtbird 325	. 2 ⊢ (𝜑 → ¬ (𝑀‘𝐴) ∈ 𝐷)
26	1, 2, 3, 6, 7, 5, 11	midbtwn 28805	. 2 ⊢ (𝜑 → (𝐴(midG‘𝐺)(𝑀‘𝐴)) ∈ (𝐴𝐼(𝑀‘𝐴)))
27	1, 2, 3, 4, 5, 11, 15, 16, 25, 26	islnoppd 28766	1 ⊢ (𝜑 → 𝐴𝑂(𝑀‘𝐴))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ∃wrex 3076 ∖ cdif 3973 class class class wbr 5166 {copab 5228 ran crn 5701 ‘cfv 6573 (class class class)co 7448 2c2 12348 Basecbs 17258 distcds 17320 TarskiGcstrkg 28453 Dim_TarskiG≥cstrkgld 28457 Itvcitv 28459 LineGclng 28460 ⟂Gcperpg 28721 midGcmid 28798 lInvGclmi 28799

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-s2 14897 df-s3 14898 df-trkgc 28474 df-trkgb 28475 df-trkgcb 28476 df-trkgld 28478 df-trkg 28479 df-cgrg 28537 df-leg 28609 df-mir 28679 df-rag 28720 df-perpg 28722 df-mid 28800 df-lmi 28801

This theorem is referenced by: trgcopyeulem 28831

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