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Theorem lmicom 28874

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 28868	. . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴))
9		lmicom.1	. . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵)
10	9	eqcomd 2745	. . . . . 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 28873	. . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))))
15	10, 14	mpbid 233	. . . . 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 877	. . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)))
20	19	ord 870	. . . . . . 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 2941	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵)
26	1, 3, 12, 21, 22, 23, 25	tglinecom 28721	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴))
27	26	breq2d 5084	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
28	27	pm5.74da 809	. . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))))
29	20, 28	mpbid 233	. . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
30	29	orrd 869	. . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))
31	30	orcomd 877	. . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵))
32		eqcom 2746	. . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴)
33	32	orbi2i 918	. . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
34	31, 33	sylib 219	. . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))
35	1, 2, 3, 4, 5, 11, 12, 13, 7, 6	islmib 28873	. . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴))))
36	17, 34, 35	mpbir2and 719	. 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵))
37	36	eqcomd 2745	1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 class class class wbr 5072 ran crn 5619 ‘cfv 6485 (class class class)co 7356 2c2 12227 Basecbs 17170 distcds 17220 TarskiGcstrkg 28513 Dim_TarskiG≥cstrkgld 28517 Itvcitv 28519 LineGclng 28520 ⟂Gcperpg 28781 midGcmid 28858 lInvGclmi 28859

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-xnn0 12502 df-z 12516 df-uz 12780 df-fz 13453 df-fzo 13600 df-hash 14284 df-word 14467 df-concat 14524 df-s1 14550 df-s2 14801 df-s3 14802 df-trkgc 28534 df-trkgb 28535 df-trkgcb 28536 df-trkgld 28538 df-trkg 28539 df-cgrg 28597 df-leg 28669 df-mir 28739 df-rag 28780 df-perpg 28782 df-mid 28860 df-lmi 28861

This theorem is referenced by: lmilmi 28875

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