Theorem mirreu 28637

Description: Any point has a unique antecedent through point inversion. Theorem 7.8 of [Schwabhauser] p. 50. (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)
mirmir.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)

Assertion

Ref	Expression
mirreu	⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵)

Distinct variable groups:   𝐵,𝑎   𝑀,𝑎   𝑃,𝑎   𝜑,𝑎

Allowed substitution hints:   𝐴(𝑎)   𝑆(𝑎)   𝐺(𝑎)   𝐼(𝑎)   𝐿(𝑎)   − (𝑎)

Proof of Theorem mirreu

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		mirmir.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	mircl 28634	. 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 28635	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
12	6	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐺 ∈ TarskiG)
13	7	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐴 ∈ 𝑃)
14		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 ∈ 𝑃)
15	1, 2, 3, 4, 5, 12, 13, 8, 14	mirmir 28635	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘𝑎) = 𝐵)
17	16	fveq2d 6821	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = (𝑀‘𝐵))
18	15, 17	eqtr3d 2768	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 = (𝑀‘𝐵))
19	18	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵)))
20	19	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵)))
21		fveqeq2 6826	. . 3 ⊢ (𝑎 = (𝑀‘𝐵) → ((𝑀‘𝑎) = 𝐵 ↔ (𝑀‘(𝑀‘𝐵)) = 𝐵))
22	21	eqreu 3683	. 2 ⊢ (((𝑀‘𝐵) ∈ 𝑃 ∧ (𝑀‘(𝑀‘𝐵)) = 𝐵 ∧ ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵))) → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵)
23	10, 11, 20, 22	syl3anc 1373	1 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃!wreu 3344  ‘cfv 6476  Basecbs 17115  distcds 17165  TarskiGcstrkg 28400  Itvcitv 28406  LineGclng 28407  pInvGcmir 28625

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 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-trkgc 28421  df-trkgb 28422  df-trkgcb 28423  df-trkg 28426  df-mir 28626

This theorem is referenced by: (None)