Theorem mircgr 28481

Description: Property of the image by the point inversion function. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 3-Jun-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
mircgr	⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))

Proof of Theorem mircgr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		mirval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		mirval.s	. . . . 5 ⊢ 𝑆 = (pInvG‘𝐺)
6		mirval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirfv.m	. . . . 5 ⊢ 𝑀 = (𝑆‘𝐴)
9		mirfv.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mirfv 28480	. . . 4 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
11	1, 2, 3, 6, 9, 7	mirreu3 28478	. . . . 5 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
12		riotacl2 7399	. . . . 5 ⊢ (∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
14	10, 13	eqeltrd 2829	. . 3 ⊢ (𝜑 → (𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))})
15		oveq2 7434	. . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 − 𝑧) = (𝐴 − (𝑀‘𝐵)))
16	15	eqeq1d 2730	. . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵)))
17		oveq1 7433	. . . . . 6 ⊢ (𝑧 = (𝑀‘𝐵) → (𝑧𝐼𝐵) = ((𝑀‘𝐵)𝐼𝐵))
18	17	eleq2d 2815	. . . . 5 ⊢ (𝑧 = (𝑀‘𝐵) → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵)))
19	16, 18	anbi12d 630	. . . 4 ⊢ (𝑧 = (𝑀‘𝐵) → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))))
20	19	elrab 3684	. . 3 ⊢ ((𝑀‘𝐵) ∈ {𝑧 ∈ 𝑃 ∣ ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))} ↔ ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))))
21	14, 20	sylib 217	. 2 ⊢ (𝜑 → ((𝑀‘𝐵) ∈ 𝑃 ∧ ((𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵) ∧ 𝐴 ∈ ((𝑀‘𝐵)𝐼𝐵))))
22	21	simprld 770	1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐵)) = (𝐴 − 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃!wreu 3372  {crab 3430  ‘cfv 6553  ℩crio 7381  (class class class)co 7426  Basecbs 17187  distcds 17249  TarskiGcstrkg 28251  Itvcitv 28257  LineGclng 28258  pInvGcmir 28476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-trkgc 28272  df-trkgb 28273  df-trkgcb 28274  df-trkg 28277  df-mir 28477

This theorem is referenced by:  mirmir  28486  miriso  28494  mirmir2  28498  mircgrextend  28506  mirtrcgr  28507  mirauto  28508  miduniq  28509  krippenlem  28514  ragcol  28523  ragflat  28528  ragcgr  28531  footexALT  28542  footexlem2  28544  colperpexlem1  28554  colperpexlem3  28556  mideulem2  28558  opphllem  28559  midcgr  28604  lmiisolem  28620