Theorem mirfv 28829

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 28828	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
10	1, 9	eqtrid 2809	. 2 ⊢ (𝜑 → 𝑀 = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
11		simplr 778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → 𝑦 = 𝐵)
12	11	oveq2d 7412	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 − 𝑦) = (𝐴 − 𝐵))
13	12	eqeq2d 2773	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → ((𝐴 − 𝑧) = (𝐴 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝐵)))
14	11	oveq2d 7412	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
15	14	eleq2d 2848	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
16	13, 15	anbi12d 641	. . 3 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
17	16	riotabidva 7372	. 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 7357	. . 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 6983	1 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ↦ cmpt 5181  ‘cfv 6521  ℩crio 7352  (class class class)co 7396  Basecbs 17245  distcds 17295  TarskiGcstrkg 28596  Itvcitv 28602  LineGclng 28603  pInvGcmir 28825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-mir 28826

This theorem is referenced by:  mircgr  28830  mirbtwn  28831  ismir  28832  mirf  28833  mireq  28838