Theorem mirval 28631

Description: Value of the point inversion function 𝑆. Definition 7.5 of [Schwabhauser] p. 49. (Contributed by Thierry Arnoux, 30-May-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)

Assertion

Ref	Expression
mirval	⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))

Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐺,𝑧   𝑦,𝐼,𝑧   𝑦,𝑃,𝑧   𝜑,𝑦,𝑧   𝑦, − ,𝑧

Allowed substitution hints:   𝑆(𝑦,𝑧)   𝐿(𝑦,𝑧)

Proof of Theorem mirval

Dummy variables 𝑥 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
2		df-mir 28629	. . . 4 ⊢ pInvG = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (℩𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))))))
3		fveq2 6822	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		mirval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
5	3, 4	eqtr4di 2784	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
7		mirval.d	. . . . . . . . . . 11 ⊢ − = (dist‘𝐺)
8	6, 7	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = − )
9	8	oveqd 7363	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 − 𝑧))
10	8	oveqd 7363	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 − 𝑦))
11	9, 10	eqeq12d 2747	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 − 𝑧) = (𝑥 − 𝑦)))
12		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
13		mirval.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
14	12, 13	eqtr4di 2784	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
15	14	oveqd 7363	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
16	15	eleq2d 2817	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
17	11, 16	anbi12d 632	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)) ↔ ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
18	5, 17	riotaeqbidv 7306	. . . . . 6 ⊢ (𝑔 = 𝐺 → (℩𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))) = (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))
19	5, 18	mpteq12dv 5178	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦 ∈ (Base‘𝑔) ↦ (℩𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))))
20	5, 19	mpteq12dv 5178	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (Base‘𝑔) ↦ (𝑦 ∈ (Base‘𝑔) ↦ (℩𝑧 ∈ (Base‘𝑔)((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ∧ 𝑥 ∈ (𝑧(Itv‘𝑔)𝑦))))) = (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
21		mirval.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
22	21	elexd 3460	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
23	4	fvexi 6836	. . . . . 6 ⊢ 𝑃 ∈ V
24	23	mptex 7157	. . . . 5 ⊢ (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V
25	24	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))) ∈ V)
26	2, 20, 22, 25	fvmptd3 6952	. . 3 ⊢ (𝜑 → (pInvG‘𝐺) = (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
27	1, 26	eqtrid 2778	. 2 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
28		simpll 766	. . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → 𝑥 = 𝐴)
29	28	oveq1d 7361	. . . . . . 7 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 − 𝑧) = (𝐴 − 𝑧))
30	28	oveq1d 7361	. . . . . . 7 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 − 𝑦) = (𝐴 − 𝑦))
31	29, 30	eqeq12d 2747	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → ((𝑥 − 𝑧) = (𝑥 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝑦)))
32	28	eleq1d 2816	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
33	31, 32	anbi12d 632	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
34	33	riotabidva 7322	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
35	34	mpteq2dva 5184	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
36	35	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)))) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))
37		mirval.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
38	23	mptex 7157	. . 3 ⊢ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V
39	38	a1i 11	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))) ∈ V)
40	27, 36, 37, 39	fvmptd 6936	1 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ↦ cmpt 5172  ‘cfv 6481  ℩crio 7302  (class class class)co 7346  Basecbs 17117  distcds 17167  TarskiGcstrkg 28403  Itvcitv 28409  LineGclng 28410  pInvGcmir 28628

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 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

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-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  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-mir 28629

This theorem is referenced by:  mirfv  28632  mirf  28636