Theorem mirfv 26594

Description: Value of the point inversion function 𝑀. Definition 7.5 of [Schwabhauser] p. 49. (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)
mirfv.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
mirfv	⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐺   𝑧,𝑀   𝑧,𝐼   𝑧,𝑃   𝜑,𝑧   𝑧, −

Allowed substitution hints:   𝑆(𝑧)   𝐿(𝑧)

Proof of Theorem mirfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
2		mirval.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		mirval.d	. . . 4 ⊢ − = (dist‘𝐺)
4		mirval.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		mirval.l	. . . 4 ⊢ 𝐿 = (LineG‘𝐺)
6		mirval.s	. . . 4 ⊢ 𝑆 = (pInvG‘𝐺)
7		mirval.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8		mirval.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
9	2, 3, 4, 5, 6, 7, 8	mirval 26593	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
10	1, 9	syl5eq 2785	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → 𝑦 = 𝐵)
12	11	oveq2d 7180	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 − 𝑦) = (𝐴 − 𝐵))
13	12	eqeq2d 2749	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → ((𝐴 − 𝑧) = (𝐴 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝐵)))
14	11	oveq2d 7180	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
15	14	eleq2d 2818	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
16	13, 15	anbi12d 634	. . 3 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
17	16	riotabidva 7141	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
18		mirfv.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
19		riotaex 7125	. . 3 ⊢ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V
20	19	a1i 11	. 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ V)
21	10, 17, 18, 20	fvmptd 6776	1 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2113  Vcvv 3397   ↦ cmpt 5107  ‘cfv 6333  ℩crio 7120  (class class class)co 7164  Basecbs 16579  distcds 16670  TarskiGcstrkg 26368  Itvcitv 26374  LineGclng 26375  pInvGcmir 26590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  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 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-mir 26591

This theorem is referenced by:  mircgr  26595  mirbtwn  26596  ismir  26597  mirf  26598  mireq  26603