Theorem mirreu 28672

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 28669	. 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 28670	. 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 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 ∈ 𝑃)
15	1, 2, 3, 4, 5, 12, 13, 8, 14	mirmir 28670	. . . . 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 6910	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = (𝑀‘𝐵))
18	15, 17	eqtr3d 2779	. . . 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 3146	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵)))
21		fveqeq2 6915	. . 3 ⊢ (𝑎 = (𝑀‘𝐵) → ((𝑀‘𝑎) = 𝐵 ↔ (𝑀‘(𝑀‘𝐵)) = 𝐵))
22	21	eqreu 3735	. 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 1540   ∈ wcel 2108  ∀wral 3061  ∃!wreu 3378  ‘cfv 6561  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441  LineGclng 28442  pInvGcmir 28660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-trkgc 28456  df-trkgb 28457  df-trkgcb 28458  df-trkg 28461  df-mir 28661

This theorem is referenced by: (None)