Theorem mirfv 28740

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 28739	. . 3 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
10	1, 9	eqtrid 2784	. 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 7384	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 − 𝑦) = (𝐴 − 𝐵))
13	12	eqeq2d 2748	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → ((𝐴 − 𝑧) = (𝐴 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝐵)))
14	11	oveq2d 7384	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝑧𝐼𝑦) = (𝑧𝐼𝐵))
15	14	eleq2d 2823	. . . 4 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (𝐴 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝐵)))
16	13, 15	anbi12d 633	. . 3 ⊢ (((𝜑 ∧ 𝑦 = 𝐵) ∧ 𝑧 ∈ 𝑃) → (((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
17	16	riotabidva 7344	. 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 7329	. . 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 6957	1 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ↦ cmpt 5181  ‘cfv 6500  ℩crio 7324  (class class class)co 7368  Basecbs 17148  distcds 17198  TarskiGcstrkg 28511  Itvcitv 28517  LineGclng 28518  pInvGcmir 28736

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-mir 28737

This theorem is referenced by:  mircgr  28741  mirbtwn  28742  ismir  28743  mirf  28744  mireq  28749