Theorem axprecex 6954

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 6994.

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 6905	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 2312	. . . 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 3768	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐴))
5		oveq1 5519	. . . . . . 7 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2048	. . . . . 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 2327	. . . 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 6896	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 3771	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
12		ltresr 6915	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
13	11, 12	bitri 173	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
14		recexgt0sr 6858	. . . . 5 ⊢ (0R <R 𝑦 → ∃𝑧 ∈ R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15		opelreal 6904	. . . . . . . . . 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 3771	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
18		ltresr 6915	. . . . . . . . . . . . 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 6914	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
22	21	eqeq1d 2048	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23		df-1 6897	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2050	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2040	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 6836	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 6835	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 3976	. . . . . . . . . . . . . . 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 848	. . . . . . . . . . . . 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 3768	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (0 <ℝ 𝑥 ↔ 0 <ℝ ⟨𝑧, 0R⟩))
37		oveq2 5520	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
38	37	eqeq1d 2048	. . . . . . . . . 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 2656	. . . . . . . 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 2432	. . . . 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 2586	. 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 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  (class class class)co 5512  Rcnr 6395  0_Rc0r 6396  1_Rc1r 6397   ·_R cmr 6400   <_R cltr 6401  ℝcr 6888  0cc0 6889  1c1 6890   <_ℝ cltrr 6893   · cmul 6894

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-imp 6567  df-iltp 6568  df-enr 6811  df-nr 6812  df-plr 6813  df-mr 6814  df-ltr 6815  df-0r 6816  df-1r 6817  df-m1r 6818  df-c 6895  df-0 6896  df-1 6897  df-r 6899  df-mul 6901  df-lt 6902

This theorem is referenced by:  rereceu  6963  recriota  6964