Theorem mirval 28913

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 28911	. . . 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 6883	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		mirval.p	. . . . . 6 ⊢ 𝑃 = (Base‘𝐺)
5	3, 4	eqtr4di 2816	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
6		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
7		mirval.d	. . . . . . . . . . 11 ⊢ − = (dist‘𝐺)
8	6, 7	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = − )
9	8	oveqd 7429	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑧) = (𝑥 − 𝑧))
10	8	oveqd 7429	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑥(dist‘𝑔)𝑦) = (𝑥 − 𝑦))
11	9, 10	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ((𝑥(dist‘𝑔)𝑧) = (𝑥(dist‘𝑔)𝑦) ↔ (𝑥 − 𝑧) = (𝑥 − 𝑦)))
12		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
13		mirval.i	. . . . . . . . . . 11 ⊢ 𝐼 = (Itv‘𝐺)
14	12, 13	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
15	14	oveqd 7429	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (𝑧(Itv‘𝑔)𝑦) = (𝑧𝐼𝑦))
16	15	eleq2d 2849	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑥 ∈ (𝑧(Itv‘𝑔)𝑦) ↔ 𝑥 ∈ (𝑧𝐼𝑦)))
17	11, 16	anbi12d 643	. . . . . . 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 7372	. . . . . 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 5199	. . . . 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 5199	. . . 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 3478	. . . 4 ⊢ (𝜑 → 𝐺 ∈ V)
23	4	fvexi 6897	. . . . . 6 ⊢ 𝑃 ∈ V
24	23	mptex 7223	. . . . 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 7015	. . 3 ⊢ (𝜑 → (pInvG‘𝐺) = (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
27	1, 26	eqtrid 2810	. 2 ⊢ (𝜑 → 𝑆 = (𝑥 ∈ 𝑃 ↦ (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))))))
28		simpll 778	. . . . . . . 8 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → 𝑥 = 𝐴)
29	28	oveq1d 7427	. . . . . . 7 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 − 𝑧) = (𝐴 − 𝑧))
30	28	oveq1d 7427	. . . . . . 7 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 − 𝑦) = (𝐴 − 𝑦))
31	29, 30	eqeq12d 2779	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → ((𝑥 − 𝑧) = (𝑥 − 𝑦) ↔ (𝐴 − 𝑧) = (𝐴 − 𝑦)))
32	28	eleq1d 2848	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝐴 ∈ (𝑧𝐼𝑦)))
33	31, 32	anbi12d 643	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) ∧ 𝑧 ∈ 𝑃) → (((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦)) ↔ ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
34	33	riotabidva 7388	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 ∈ 𝑃) → (℩𝑧 ∈ 𝑃 ((𝑥 − 𝑧) = (𝑥 − 𝑦) ∧ 𝑥 ∈ (𝑧𝐼𝑦))) = (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦))))
35	34	mpteq2dva 5205	. . 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 486	. 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 7223	. . 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 6999	1 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑦 ∈ 𝑃 ↦ (℩𝑧 ∈ 𝑃 ((𝐴 − 𝑧) = (𝐴 − 𝑦) ∧ 𝐴 ∈ (𝑧𝐼𝑦)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5193  ‘cfv 6538  ℩crio 7368  (class class class)co 7412  Basecbs 17270  distcds 17320  TarskiGcstrkg 28677  Itvcitv 28683  LineGclng 28684  pInvGcmir 28910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-mir 28911

This theorem is referenced by:  mirfv  28914  mirf  28918