Theorem mirfv 25750

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 25749	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
10	1, 9	syl5eq 2806	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11		simplr 809	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → 𝑦 = 𝐵)
12	11	oveq2d 6829	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 − 𝑦) = (𝐴 − 𝐵))
13	12	eqeq2d 2770	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → ((𝐴 − 𝑧) = (𝐴 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝐵)))
14	11	oveq2d 6829	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
15	14	eleq2d 2825	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
16	13, 15	anbi12d 749	. . 3 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
17	16	riotabidva 6790	. 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 6778	. . 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 6450	1 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ↦ cmpt 4881  ‘cfv 6049  ℩crio 6773  (class class class)co 6813  Basecbs 16059  distcds 16152  TarskiGcstrkg 25528  Itvcitv 25534  LineGclng 25535  pInvGcmir 25746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-mir 25747

This theorem is referenced by:  mircgr  25751  mirbtwn  25752  ismir  25753  mirf  25754  mireq  25759