Theorem mirreu 29016

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 29013	. 2 ⊢ (𝜑 → (𝑀‘𝐵) ∈ 𝑃)
11	1, 2, 3, 4, 5, 6, 7, 8, 9	mirmir 29014	. 2 ⊢ (𝜑 → (𝑀‘(𝑀‘𝐵)) = 𝐵)
12	6	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐺 ∈ TarskiG)
13	7	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝐴 ∈ 𝑃)
14		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 ∈ 𝑃)
15	1, 2, 3, 4, 5, 12, 13, 8, 14	mirmir 29014	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = 𝑎)
16		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘𝑎) = 𝐵)
17	16	fveq2d 6883	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → (𝑀‘(𝑀‘𝑎)) = (𝑀‘𝐵))
18	15, 17	eqtr3d 2797	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝑃) ∧ (𝑀‘𝑎) = 𝐵) → 𝑎 = (𝑀‘𝐵))
19	18	ex 418	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑃) → ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵)))
20	19	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝑃 ((𝑀‘𝑎) = 𝐵 → 𝑎 = (𝑀‘𝐵)))
21		fveqeq2 6888	. . 3 ⊢ (𝑎 = (𝑀‘𝐵) → ((𝑀‘𝑎) = 𝐵 ↔ (𝑀‘(𝑀‘𝐵)) = 𝐵))
22	21	eqreu 3687	. 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 1398	1 ⊢ (𝜑 → ∃!𝑎 ∈ 𝑃 (𝑀‘𝑎) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363  ‘cfv 6533  Basecbs 17302  distcds 17352  TarskiGcstrkg 28769  Itvcitv 28775  LineGclng 28776  pInvGcmir 29004

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-trkgc 28790  df-trkgb 28791  df-trkgcb 28792  df-trkg 28795  df-mir 29005

This theorem is used by: (None)