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Theorem mirauto 28710

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 28686	. . . . 5 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
11		mirauto.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
12	10, 11	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
13	7, 12	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
14		eqid 2740	. . 3 ⊢ (𝑆‘𝑋) = (𝑆‘𝑋)
15		mirauto.y	. . . 4 ⊢ 𝑌 = (𝑀‘𝐵)
16		mirauto.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
17	10, 16	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
18	15, 17	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
19		mirauto.z	. . . 4 ⊢ 𝑍 = (𝑀‘𝐶)
20		mirauto.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
21	10, 20	ffvelcdmd 7119	. . . 4 ⊢ (𝜑 → (𝑀‘𝐶) ∈ 𝑃)
22	19, 21	eqeltrid 2848	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
23		mirauto.4	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) = 𝐶)
24	23, 20	eqeltrd 2844	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐴)‘𝐵) ∈ 𝑃)
25		eqid 2740	. . . . . 6 ⊢ (𝑆‘𝐴) = (𝑆‘𝐴)
26	1, 2, 3, 4, 5, 6, 11, 25, 16	mircgr 28683	. . . . 5 ⊢ (𝜑 → (𝐴 − ((𝑆‘𝐴)‘𝐵)) = (𝐴 − 𝐵))
27	1, 2, 3, 4, 5, 6, 8, 9, 11, 24, 11, 16, 26	mircgrs 28699	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
28	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑋 = (𝑀‘𝐴))
29	23	fveq2d 6924	. . . . . 6 ⊢ (𝜑 → (𝑀‘((𝑆‘𝐴)‘𝐵)) = (𝑀‘𝐶))
30	19, 29	eqtr4id 2799	. . . . 5 ⊢ (𝜑 → 𝑍 = (𝑀‘((𝑆‘𝐴)‘𝐵)))
31	28, 30	oveq12d 7466	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑍) = ((𝑀‘𝐴) − (𝑀‘((𝑆‘𝐴)‘𝐵))))
32	7, 15	oveq12i 7460	. . . . 5 ⊢ (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵))
33	32	a1i 11	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) = ((𝑀‘𝐴) − (𝑀‘𝐵)))
34	27, 31, 33	3eqtr4d 2790	. . 3 ⊢ (𝜑 → (𝑋 − 𝑍) = (𝑋 − 𝑌))
35	1, 2, 3, 4, 5, 6, 11, 25, 16	mirbtwn 28684	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (((𝑆‘𝐴)‘𝐵)𝐼𝐵))
36	23	oveq1d 7463	. . . . . 6 ⊢ (𝜑 → (((𝑆‘𝐴)‘𝐵)𝐼𝐵) = (𝐶𝐼𝐵))
37	35, 36	eleqtrd 2846	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))
38	1, 2, 3, 4, 5, 6, 8, 9, 20, 11, 16, 37	mirbtwni 28697	. . . 4 ⊢ (𝜑 → (𝑀‘𝐴) ∈ ((𝑀‘𝐶)𝐼(𝑀‘𝐵)))
39	19, 15	oveq12i 7460	. . . 4 ⊢ (𝑍𝐼𝑌) = ((𝑀‘𝐶)𝐼(𝑀‘𝐵))
40	38, 7, 39	3eltr4g 2861	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑍𝐼𝑌))
41	1, 2, 3, 4, 5, 6, 13, 14, 18, 22, 34, 40	ismir 28685	. 2 ⊢ (𝜑 → 𝑍 = ((𝑆‘𝑋)‘𝑌))
42	41	eqcomd 2746	1 ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 distcds 17320 TarskiGcstrkg 28453 Itvcitv 28459 LineGclng 28460 pInvGcmir 28678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-s2 14897 df-s3 14898 df-trkgc 28474 df-trkgb 28475 df-trkgcb 28476 df-trkg 28479 df-cgrg 28537 df-mir 28679

This theorem is referenced by: miduniq2 28713 krippenlem 28716

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