Theorem mirf 28891

Description: Point inversion as function. (Contributed by Thierry Arnoux, 30-May-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)
mirval.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirfv.m	⊢ 𝑀 = (𝑆‘𝐴)

Assertion

Ref	Expression
mirf	⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Proof of Theorem mirf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 7361	. . 3 ⊢ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V
2	1	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) ∈ V)
3		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
4		mirval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
5		mirval.d	. . . 4 ⊢ − = (dist‘𝐺)
6		mirval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
7		mirval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
10		mirval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11	4, 5, 6, 7, 8, 9, 10	mirval 28886	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
12	3, 11	eqtrid 2812	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
13	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐺 ∈ TarskiG)
14	10	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐴 ∈ 𝑃)
15		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	4, 5, 6, 7, 8, 13, 14, 3, 15	mirfv 28887	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
17	4, 5, 6, 13, 15, 14	mirreu3 28885	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18		riotacl 7374	. . . 4 ⊢ (∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
19	17, 18	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
20	16, 19	eqeltrd 2865	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) ∈ 𝑃)
21	2, 12, 20	fmpt2d 7110	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃!wreu 3368  Vcvv 3457   ↦ cmpt 5186  ⟶wf 6521  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  Basecbs 17259  distcds 17309  TarskiGcstrkg 28654  Itvcitv 28660  LineGclng 28661  pInvGcmir 28883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-trkgc 28675  df-trkgb 28676  df-trkgcb 28677  df-trkg 28680  df-mir 28884

This theorem is referenced by:  mircl  28892  mirf1o  28900  mirbtwni  28902  mirbtwnb  28903  mirauto  28915  miduniq2  28918  krippenlem  28921