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Theorem mirauto 27045

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 27021	. . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
11		mirauto.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12	10, 11	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
13	7, 12	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14		eqid 2738	. . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋)
15		mirauto.y	. . . 4 ⊢ 𝑌 = (𝑀‘𝐵)
16		mirauto.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
17	10, 16	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
18	15, 17	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
19		mirauto.z	. . . 4 ⊢ 𝑍 = (𝑀‘𝐶)
20		mirauto.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
21	10, 20	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
22	19, 21	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
23		mirauto.4	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶)
24	23, 20	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃)
25		eqid 2738	. . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
26	1, 2, 3, 4, 5, 6, 11, 25, 16	mircgr 27018	. . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵))
27	1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26	mircgrs 27034	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
28	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴))
29	23	fveq2d 6778	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶))
30	19, 29	eqtr4id 2797	. . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵)))
31	28, 30	oveq12d 7293	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))))
32	7, 15	oveq12i 7287	. . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))
33	32	a1i 11	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
34	27, 31, 33	3eqtr4d 2788	. . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌))
35	1, 2, 3, 4, 5, 6, 11, 25, 16	mirbtwn 27019	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵))
36	23	oveq1d 7290	. . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵))
37	35, 36	eleqtrd 2841	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))
38	1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37	mirbtwni 27032	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵)))
39	19, 15	oveq12i 7287	. . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵))
40	38, 7, 39	3eltr4g 2856	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))
41	1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40	ismir 27020	. 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌))
42	41	eqcomd 2744	1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 distcds 16971 TarskiGcstrkg 26788 Itvcitv 26794 LineGclng 26795 pInvGcmir 27013

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-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-trkg 26814 df-cgrg 26872 df-mir 27014

This theorem is referenced by: miduniq2 27048 krippenlem 27051

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