Theorem ismir 28637

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 28634	. 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 28632	. . . 4 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝐵) ∧ 𝐴 ∈ (𝑧𝐼𝐵)))
15		oveq2 7354	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝐴 − 𝑧) = (𝐴 − 𝐶))
16	15	eqeq1d 2733	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐴 − 𝑧) = (𝐴 − 𝐵) ↔ (𝐴 − 𝐶) = (𝐴 − 𝐵)))
17		oveq1 7353	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧𝐼𝐵) = (𝐶𝐼𝐵))
18	17	eleq2d 2817	. . . . . 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 7328	. . . 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 2767	1 ⊢ (𝜑 → 𝐶 = (𝑀‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!wreu 3344  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  Basecbs 17120  distcds 17170  TarskiGcstrkg 28405  Itvcitv 28411  LineGclng 28412  pInvGcmir 28630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-trkgc 28426  df-trkgb 28427  df-trkgcb 28428  df-trkg 28431  df-mir 28631

This theorem is referenced by:  mirmir  28640  mireq  28643  mirinv  28644  miriso  28648  mirmir2  28652  mirauto  28662  colmid  28666  krippenlem  28668  midexlem  28670  mideulem2  28712  opphllem  28713  midcom  28760  trgcopyeulem  28783