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Theorem lmicom 27149

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 27143	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
9		lmicom.1	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵)
10	9	eqcomd 2744	. . . . . 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 27148	. . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
15	10, 14	mpbid 231	. . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))
16	15	simpld 495	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷)
17	8, 16	eqeltrrd 2840	. . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷)
18	15	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
19	18	orcomd 868	. . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
20	19	ord 861	. . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
21	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
22	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃)
23	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
24		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵)
25	24	neqned 2950	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵)
26	1, 3, 12, 21, 22, 23, 25	tglinecom 26996	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
27	26	breq2d 5086	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
28	27	pm5.74da 801	. . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))))
29	20, 28	mpbid 231	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
30	29	orrd 860	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
31	30	orcomd 868	. . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵))
32		eqcom 2745	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
33	32	orbi2i 910	. . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
34	31, 33	sylib 217	. . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
35	1, 2, 3, 4, 5, 11, 12, 13, 7, 6	islmib 27148	. . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))))
36	17, 34, 35	mpbir2and 710	. 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵))
37	36	eqcomd 2744	1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ran crn 5590 ‘cfv 6433 (class class class)co 7275 2c2 12028 Basecbs 16912 distcds 16971 TarskiGcstrkg 26788 Dim_TarskiG≥cstrkgld 26792 Itvcitv 26794 LineGclng 26795 ⟂Gcperpg 27056 midGcmid 27133 lInvGclmi 27134

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-s2 14561 df-s3 14562 df-trkgc 26809 df-trkgb 26810 df-trkgcb 26811 df-trkgld 26813 df-trkg 26814 df-cgrg 26872 df-leg 26944 df-mir 27014 df-rag 27055 df-perpg 27057 df-mid 27135 df-lmi 27136

This theorem is referenced by: lmilmi 27150

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