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Theorem mirauto 28707

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 28683	. . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
11		mirauto.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12	10, 11	ffvelcdmd 7105	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
13	7, 12	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14		eqid 2735	. . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋)
15		mirauto.y	. . . 4 ⊢ 𝑌 = (𝑀‘𝐵)
16		mirauto.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
17	10, 16	ffvelcdmd 7105	. . . 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	ffvelcdmd 7105	. . . 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 2735	. . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
26	1, 2, 3, 4, 5, 6, 11, 25, 16	mircgr 28680	. . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵))
27	1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26	mircgrs 28696	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
28	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴))
29	23	fveq2d 6911	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶))
30	19, 29	eqtr4id 2794	. . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵)))
31	28, 30	oveq12d 7449	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))))
32	7, 15	oveq12i 7443	. . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))
33	32	a1i 11	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
34	27, 31, 33	3eqtr4d 2785	. . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌))
35	1, 2, 3, 4, 5, 6, 11, 25, 16	mirbtwn 28681	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵))
36	23	oveq1d 7446	. . . . . 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 28694	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵)))
39	19, 15	oveq12i 7443	. . . 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 28682	. 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌))
42	41	eqcomd 2741	1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 distcds 17307 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 pInvGcmir 28675

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-s2 14884 df-s3 14885 df-trkgc 28471 df-trkgb 28472 df-trkgcb 28473 df-trkg 28476 df-cgrg 28534 df-mir 28676

This theorem is referenced by: miduniq2 28710 krippenlem 28713

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