Theorem ismir 28681

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)
ismir.1	⊢ (𝜑 → 𝐶 ∈ 𝑃)
ismir.2	⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵))
ismir.3	⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))

Assertion

Ref	Expression
ismir	⊢ (𝜑 → 𝐶 = (𝑀‘𝐵))

Proof of Theorem ismir

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirval.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. . 3 ⊢ − = (dist‘𝐺)
3		mirval.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
6		mirval.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
7		mirval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8		mirfv.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐴)
9		mirfv.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mirfv 28678	. 2 ⊢ (𝜑 → (𝑀‘𝐵) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))))
11		ismir.2	. . 3 ⊢ (𝜑 → (𝐴 − 𝐶) = (𝐴 − 𝐵))
12		ismir.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐵))
13		ismir.1	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
14	1, 2, 3, 6, 9, 7	mirreu3 28676	. . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15		oveq2 7438	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 − 𝑧) = (𝐴 − 𝐶))
16	15	eqeq1d 2736	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − 𝐶) = (𝐴 − 𝐵)))
17		oveq1 7437	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
18	17	eleq2d 2824	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐴 ∈ (𝑧𝐼𝐵) ↔ 𝐴 ∈ (𝐶𝐼𝐵)))
19	16, 18	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝐶 → (((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)) ↔ ((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵))))
20	19	riota2 7412	. . . 4 ⊢ ((𝐶 ∈ 𝑃 ∧ ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) → (((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
21	13, 14, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (((𝐴 − 𝐶) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝐶𝐼𝐵)) ↔ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶))
22	11, 12, 21	mpbi2and 712	. 2 ⊢ (𝜑 → (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵))) = 𝐶)
23	10, 22	eqtr2d 2775	1 ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃!wreu 3375  ‘cfv 6562  ℩crio 7386  (class class class)co 7430  Basecbs 17244  distcds 17306  TarskiGcstrkg 28449  Itvcitv 28455  LineGclng 28456  pInvGcmir 28674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-trkgc 28470  df-trkgb 28471  df-trkgcb 28472  df-trkg 28475  df-mir 28675

This theorem is referenced by:  mirmir  28684  mireq  28687  mirinv  28688  miriso  28692  mirmir2  28696  mirauto  28706  colmid  28710  krippenlem  28712  midexlem  28714  mideulem2  28756  opphllem  28757  midcom  28804  trgcopyeulem  28827