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Theorem lmicom 26833

Description: The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
ismid.p	⊢ 𝑃 = (Base‘𝐺)
ismid.d	⊢ − = (dist‘𝐺)
ismid.i	⊢ 𝐼 = (Itv‘𝐺)
ismid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ismid.1	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmif.m	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l	⊢ 𝐿 = (LineG‘𝐺)
lmif.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lmicl.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
islmib.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
lmicom.1	⊢ (𝜑 → (𝑀‘𝐴) = 𝐵)

**Assertion**
Ref	Expression
lmicom	⊢ (𝜑 → (𝑀‘𝐵) = 𝐴)

**Proof of Theorem lmicom**
Step	Hyp	Ref	Expression
1		ismid.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		ismid.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		ismid.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		ismid.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		ismid.1	. . . . 5 ⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
6		lmicl.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		islmib.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	1, 2, 3, 4, 5, 6, 7	midcom 26827	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
9		lmicom.1	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵)
10	9	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴))
11		lmif.m	. . . . . . 7 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
12		lmif.l	. . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺)
13		lmif.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
14	1, 2, 3, 4, 5, 11, 12, 13, 6, 7	islmib 26832	. . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
15	10, 14	mpbid 235	. . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
16	15	simpld 498	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷)
17	8, 16	eqeltrrd 2832	. . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷)
18	15	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
19	18	orcomd 871	. . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
20	19	ord 864	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
21	4	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
22	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
23	7	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
24		simpr 488	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
25	24	neqned 2939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵)
26	1, 3, 12, 21, 22, 23, 25	tglinecom 26680	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
27	26	breq2d 5051	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
28	27	pm5.74da 804	. . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))))
29	20, 28	mpbid 235	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
30	29	orrd 863	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
31	30	orcomd 871	. . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵))
32		eqcom 2743	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
33	32	orbi2i 913	. . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
34	31, 33	sylib 221	. . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
35	1, 2, 3, 4, 5, 11, 12, 13, 7, 6	islmib 26832	. . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))))
36	17, 34, 35	mpbir2and 713	. 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵))
37	36	eqcomd 2742	1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 ran crn 5537 ‘cfv 6358 (class class class)co 7191 2c2 11850 Basecbs 16666 distcds 16758 TarskiGcstrkg 26475 Dim_TarskiG≥cstrkgld 26479 Itvcitv 26481 LineGclng 26482 ⟂Gcperpg 26740 midGcmid 26817 lInvGclmi 26818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-map 8488 df-pm 8489 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-dju 9482 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-fz 13061 df-fzo 13204 df-hash 13862 df-word 14035 df-concat 14091 df-s1 14118 df-s2 14378 df-s3 14379 df-trkgc 26493 df-trkgb 26494 df-trkgcb 26495 df-trkgld 26497 df-trkg 26498 df-cgrg 26556 df-leg 26628 df-mir 26698 df-rag 26739 df-perpg 26741 df-mid 26819 df-lmi 26820

This theorem is referenced by: lmilmi 26834

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