Theorem axprecex 7942

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 7984.

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 7890	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 2478	. . . 4 ⊢ (∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
3	1, 2	bitri 184	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4		breq2 4034	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐴))
5		oveq1 5926	. . . . . . 7 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2202	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
7	6	anbi2d 464	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)))
8	7	rexbidv 2495	. . . 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 234	. . 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 7881	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 4037	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
12		ltresr 7901	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
13	11, 12	bitri 184	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
14		recexgt0sr 7835	. . . . 5 ⊢ (0R <R 𝑦 → ∃𝑧 ∈ R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15		opelreal 7889	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
16	15	anbi1i 458	. . . . . . . . 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 4037	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
18		ltresr 7901	. . . . . . . . . . . . 13 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 0R <R 𝑧)
19	17, 18	bitri 184	. . . . . . . . . . . 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 7900	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
22	21	eqeq1d 2202	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23		df-1 7882	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2204	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2193	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 7813	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 7812	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 4269	. . . . . . . . . . . . . . 15 ⊢ ((1R ∈ R ∧ 0R ∈ R) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
29	26, 27, 28	mp2an 426	. . . . . . . . . . . . . 14 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
30	25, 29	mpbiran2 943	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
31	24, 30	bitri 184	. . . . . . . . . . . 12 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
32	22, 31	bitrdi 196	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
33	20, 32	anbi12d 473	. . . . . . . . . 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 452	. . . . . . . . 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	bitrid 192	. . . . . . . 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 4034	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (0 <ℝ 𝑥 ↔ 0 <ℝ ⟨𝑧, 0R⟩))
37		oveq2 5927	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
38	37	eqeq1d 2202	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
39	36, 38	anbi12d 473	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <ℝ ⟨𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
40	39	rspcev 2865	. . . . . . . 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	biimtrrdi 164	. . . . . . 7 ⊢ (𝑦 ∈ R → ((𝑧 ∈ R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
42	41	expd 258	. . . . . 6 ⊢ (𝑦 ∈ R → (𝑧 ∈ R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
43	42	rexlimdv 2610	. . . . 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 32	. . . 4 ⊢ (𝑦 ∈ R → (0R <R 𝑦 → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
45	13, 44	biimtrid 152	. . 3 ⊢ (𝑦 ∈ R → (0 <ℝ ⟨𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
46	3, 9, 45	gencl 2792	. 2 ⊢ (𝐴 ∈ ℝ → (0 <ℝ 𝐴 → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)))
47	46	imp 124	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∃wrex 2473  ⟨cop 3622   class class class wbr 4030  (class class class)co 5919  Rcnr 7359  0_Rc0r 7360  1_Rc1r 7361   ·_R cmr 7364   <_R cltr 7365  ℝcr 7873  0cc0 7874  1c1 7875   <_ℝ cltrr 7878   · cmul 7879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-iplp 7530  df-imp 7531  df-iltp 7532  df-enr 7788  df-nr 7789  df-plr 7790  df-mr 7791  df-ltr 7792  df-0r 7793  df-1r 7794  df-m1r 7795  df-c 7880  df-0 7881  df-1 7882  df-r 7884  df-mul 7886  df-lt 7887

This theorem is referenced by:  rereceu  7951  recriota  7952