Theorem axprecex 6724

Description: Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 6753.

In treatments which assume excluded middle, the 0 <_ℝ A condition is generally replaced by A ≠ 0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axprecex	⊢ ((A ∈ ℝ ∧ 0 <ℝ A) → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1))

Distinct variable group:   x,A

Proof of Theorem axprecex

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 6687	. . . 4 ⊢ (A ∈ ℝ ↔ ∃y ∈ R ⟨y, 0R⟩ = A)
2		df-rex 2306	. . . 4 ⊢ (∃y ∈ R ⟨y, 0R⟩ = A ↔ ∃y(y ∈ R ∧ ⟨y, 0R⟩ = A))
3	1, 2	bitri 173	. . 3 ⊢ (A ∈ ℝ ↔ ∃y(y ∈ R ∧ ⟨y, 0R⟩ = A))
4		breq2 3759	. . . 4 ⊢ (⟨y, 0R⟩ = A → (0 <ℝ ⟨y, 0R⟩ ↔ 0 <ℝ A))
5		oveq1 5462	. . . . . . 7 ⊢ (⟨y, 0R⟩ = A → (⟨y, 0R⟩ · x) = (A · x))
6	5	eqeq1d 2045	. . . . . 6 ⊢ (⟨y, 0R⟩ = A → ((⟨y, 0R⟩ · x) = 1 ↔ (A · x) = 1))
7	6	anbi2d 437	. . . . 5 ⊢ (⟨y, 0R⟩ = A → ((0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1) ↔ (0 <ℝ x ∧ (A · x) = 1)))
8	7	rexbidv 2321	. . . 4 ⊢ (⟨y, 0R⟩ = A → (∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1) ↔ ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1)))
9	4, 8	imbi12d 223	. . 3 ⊢ (⟨y, 0R⟩ = A → ((0 <ℝ ⟨y, 0R⟩ → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1)) ↔ (0 <ℝ A → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1))))
10		df-0 6678	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 3762	. . . . 5 ⊢ (0 <ℝ ⟨y, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨y, 0R⟩)
12		ltresr 6696	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨y, 0R⟩ ↔ 0R <R y)
13	11, 12	bitri 173	. . . 4 ⊢ (0 <ℝ ⟨y, 0R⟩ ↔ 0R <R y)
14		recexgt0sr 6661	. . . . 5 ⊢ (0R <R y → ∃z ∈ R (0R <R z ∧ (y ·R z) = 1R))
15		opelreal 6686	. . . . . . . . . 10 ⊢ (⟨z, 0R⟩ ∈ ℝ ↔ z ∈ R)
16	15	anbi1i 431	. . . . . . . . 9 ⊢ ((⟨z, 0R⟩ ∈ ℝ ∧ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z ∈ R ∧ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)))
17	10	breq1i 3762	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨z, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨z, 0R⟩)
18		ltresr 6696	. . . . . . . . . . . . 13 ⊢ (⟨0R, 0R⟩ <ℝ ⟨z, 0R⟩ ↔ 0R <R z)
19	17, 18	bitri 173	. . . . . . . . . . . 12 ⊢ (0 <ℝ ⟨z, 0R⟩ ↔ 0R <R z)
20	19	a1i 9	. . . . . . . . . . 11 ⊢ ((y ∈ R ∧ z ∈ R) → (0 <ℝ ⟨z, 0R⟩ ↔ 0R <R z))
21		mulresr 6695	. . . . . . . . . . . . 13 ⊢ ((y ∈ R ∧ z ∈ R) → (⟨y, 0R⟩ · ⟨z, 0R⟩) = ⟨(y ·R z), 0R⟩)
22	21	eqeq1d 2045	. . . . . . . . . . . 12 ⊢ ((y ∈ R ∧ z ∈ R) → ((⟨y, 0R⟩ · ⟨z, 0R⟩) = 1 ↔ ⟨(y ·R z), 0R⟩ = 1))
23		df-1 6679	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2047	. . . . . . . . . . . . 13 ⊢ (⟨(y ·R z), 0R⟩ = 1 ↔ ⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2037	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 6639	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 6638	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 3967	. . . . . . . . . . . . . . 15 ⊢ ((1R ∈ R ∧ 0R ∈ R) → (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ ((y ·R z) = 1R ∧ 0R = 0R)))
29	26, 27, 28	mp2an 402	. . . . . . . . . . . . . 14 ⊢ (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ ((y ·R z) = 1R ∧ 0R = 0R))
30	25, 29	mpbiran2 847	. . . . . . . . . . . . 13 ⊢ (⟨(y ·R z), 0R⟩ = ⟨1R, 0R⟩ ↔ (y ·R z) = 1R)
31	24, 30	bitri 173	. . . . . . . . . . . 12 ⊢ (⟨(y ·R z), 0R⟩ = 1 ↔ (y ·R z) = 1R)
32	22, 31	syl6bb 185	. . . . . . . . . . 11 ⊢ ((y ∈ R ∧ z ∈ R) → ((⟨y, 0R⟩ · ⟨z, 0R⟩) = 1 ↔ (y ·R z) = 1R))
33	20, 32	anbi12d 442	. . . . . . . . . 10 ⊢ ((y ∈ R ∧ z ∈ R) → ((0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1) ↔ (0R <R z ∧ (y ·R z) = 1R)))
34	33	pm5.32da 425	. . . . . . . . 9 ⊢ (y ∈ R → ((z ∈ R ∧ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z ∈ R ∧ (0R <R z ∧ (y ·R z) = 1R))))
35	16, 34	syl5bb 181	. . . . . . . 8 ⊢ (y ∈ R → ((⟨z, 0R⟩ ∈ ℝ ∧ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) ↔ (z ∈ R ∧ (0R <R z ∧ (y ·R z) = 1R))))
36		breq2 3759	. . . . . . . . . 10 ⊢ (x = ⟨z, 0R⟩ → (0 <ℝ x ↔ 0 <ℝ ⟨z, 0R⟩))
37		oveq2 5463	. . . . . . . . . . 11 ⊢ (x = ⟨z, 0R⟩ → (⟨y, 0R⟩ · x) = (⟨y, 0R⟩ · ⟨z, 0R⟩))
38	37	eqeq1d 2045	. . . . . . . . . 10 ⊢ (x = ⟨z, 0R⟩ → ((⟨y, 0R⟩ · x) = 1 ↔ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1))
39	36, 38	anbi12d 442	. . . . . . . . 9 ⊢ (x = ⟨z, 0R⟩ → ((0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1) ↔ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)))
40	39	rspcev 2650	. . . . . . . 8 ⊢ ((⟨z, 0R⟩ ∈ ℝ ∧ (0 <ℝ ⟨z, 0R⟩ ∧ (⟨y, 0R⟩ · ⟨z, 0R⟩) = 1)) → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1))
41	35, 40	syl6bir 153	. . . . . . 7 ⊢ (y ∈ R → ((z ∈ R ∧ (0R <R z ∧ (y ·R z) = 1R)) → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1)))
42	41	expd 245	. . . . . 6 ⊢ (y ∈ R → (z ∈ R → ((0R <R z ∧ (y ·R z) = 1R) → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1))))
43	42	rexlimdv 2426	. . . . 5 ⊢ (y ∈ R → (∃z ∈ R (0R <R z ∧ (y ·R z) = 1R) → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1)))
44	14, 43	syl5 28	. . . 4 ⊢ (y ∈ R → (0R <R y → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1)))
45	13, 44	syl5bi 141	. . 3 ⊢ (y ∈ R → (0 <ℝ ⟨y, 0R⟩ → ∃x ∈ ℝ (0 <ℝ x ∧ (⟨y, 0R⟩ · x) = 1)))
46	3, 9, 45	gencl 2580	. 2 ⊢ (A ∈ ℝ → (0 <ℝ A → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1)))
47	46	imp 115	1 ⊢ ((A ∈ ℝ ∧ 0 <ℝ A) → ∃x ∈ ℝ (0 <ℝ x ∧ (A · x) = 1))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  ⟨cop 3370   class class class wbr 3755  (class class class)co 5455  Rcnr 6281  0_Rc0r 6282  1_Rc1r 6283   ·_R cmr 6286   <_R cltr 6287  ℝcr 6670  0cc0 6671  1c1 6672   <_ℝ cltrr 6675   · cmul 6676

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-imp 6451  df-iltp 6452  df-enr 6614  df-nr 6615  df-plr 6616  df-mr 6617  df-ltr 6618  df-0r 6619  df-1r 6620  df-m1r 6621  df-c 6677  df-0 6678  df-1 6679  df-r 6681  df-mul 6683  df-lt 6684

This theorem is referenced by: (None)