Theorem mirf 28636

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 7307	. . 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 28631	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
12	3, 11	eqtrid 2778	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
13	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐺 ∈ TarskiG)
14	10	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝐴 ∈ 𝑃)
15		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ 𝑃)
16	4, 5, 6, 7, 8, 13, 14, 3, 15	mirfv 28632	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥))))
17	4, 5, 6, 13, 15, 14	mirreu3 28630	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑥) ∧ 𝐴 ∈ (𝑧𝐼𝑥)))
18		riotacl 7320	. . . 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 2831	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑃) → (𝑀‘𝑥) ∈ 𝑃)
21	2, 12, 20	fmpt2d 7057	1 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!wreu 3344  Vcvv 3436   ↦ cmpt 5172  ⟶wf 6477  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  Basecbs 17117  distcds 17167  TarskiGcstrkg 28403  Itvcitv 28409  LineGclng 28410  pInvGcmir 28628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-trkgc 28424  df-trkgb 28425  df-trkgcb 28426  df-trkg 28429  df-mir 28629

This theorem is referenced by:  mircl  28637  mirf1o  28645  mirbtwni  28647  mirbtwnb  28648  mirauto  28660  miduniq2  28663  krippenlem  28666