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Theorem mirauto 28693

Description: Point inversion preserves point inversion. (Contributed by Thierry Arnoux, 30-Jul-2019.)

**Hypotheses**
Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirauto.m	⊢ 𝑀 = (𝑆‘𝑇)
mirauto.x	⊢ 𝑋 = (𝑀‘𝐴)
mirauto.y	⊢ 𝑌 = (𝑀‘𝐵)
mirauto.z	⊢ 𝑍 = (𝑀‘𝐶)
mirauto.0	⊢ (𝜑 → 𝑇 ∈ 𝑃)
mirauto.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirauto.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirauto.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
mirauto.4	⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶)

**Assertion**
Ref	Expression
mirauto	⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)

**Proof of Theorem mirauto**
Step	Hyp	Ref	Expression
1		mirval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. . 3 ⊢ − = (dist‘𝐺)
3		mirval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
6		mirval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirauto.x	. . . 4 ⊢ 𝑋 = (𝑀‘𝐴)
8		mirauto.0	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
9		mirauto.m	. . . . . 6 ⊢ 𝑀 = (𝑆‘𝑇)
10	1, 2, 3, 4, 5, 6, 8, 9	mirf 28669	. . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
11		mirauto.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12	10, 11	ffvelcdmd 7104	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
13	7, 12	eqeltrid 2844	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14		eqid 2736	. . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋)
15		mirauto.y	. . . 4 ⊢ 𝑌 = (𝑀‘𝐵)
16		mirauto.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
17	10, 16	ffvelcdmd 7104	. . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
18	15, 17	eqeltrid 2844	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
19		mirauto.z	. . . 4 ⊢ 𝑍 = (𝑀‘𝐶)
20		mirauto.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
21	10, 20	ffvelcdmd 7104	. . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
22	19, 21	eqeltrid 2844	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
23		mirauto.4	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶)
24	23, 20	eqeltrd 2840	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃)
25		eqid 2736	. . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
26	1, 2, 3, 4, 5, 6, 11, 25, 16	mircgr 28666	. . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵))
27	1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26	mircgrs 28682	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
28	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴))
29	23	fveq2d 6909	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶))
30	19, 29	eqtr4id 2795	. . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵)))
31	28, 30	oveq12d 7450	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))))
32	7, 15	oveq12i 7444	. . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))
33	32	a1i 11	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
34	27, 31, 33	3eqtr4d 2786	. . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌))
35	1, 2, 3, 4, 5, 6, 11, 25, 16	mirbtwn 28667	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵))
36	23	oveq1d 7447	. . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵))
37	35, 36	eleqtrd 2842	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))
38	1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37	mirbtwni 28680	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵)))
39	19, 15	oveq12i 7444	. . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵))
40	38, 7, 39	3eltr4g 2857	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))
41	1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40	ismir 28668	. 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌))
42	41	eqcomd 2742	1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 pInvGcmir 28661

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-s2 14888 df-s3 14889 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkg 28462 df-cgrg 28520 df-mir 28662

This theorem is referenced by: miduniq2 28696 krippenlem 28699

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