Theorem mirf 26454

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 7097	. . 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 26449	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
12	3, 11	syl5eq 2845	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
13	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐺 ∈ TarskiG)
14	10	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐴 ∈ 𝑃)
15		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	4, 5, 6, 7, 8, 13, 14, 3, 15	mirfv 26450	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
17	4, 5, 6, 13, 15, 14	mirreu3 26448	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18		riotacl 7110	. . . 4 ⊢ (∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
19	17, 18	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))) ∈ 𝑃)
20	16, 19	eqeltrd 2890	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) ∈ 𝑃)
21	2, 12, 20	fmpt2d 6864	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!wreu 3108  Vcvv 3441   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  ℩crio 7092  (class class class)co 7135  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231  pInvGcmir 26446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247  df-mir 26447

This theorem is referenced by:  mircl  26455  mirf1o  26463  mirbtwni  26465  mirbtwnb  26466  mirauto  26478  miduniq2  26481  krippenlem  26484