Theorem axprecex 7947

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 7989.

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 7895	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 2481	. . . 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 4037	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐴))
5		oveq1 5929	. . . . . . 7 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2205	. . . . . 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 2498	. . . 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 7886	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 4040	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
12		ltresr 7906	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
13	11, 12	bitri 184	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
14		recexgt0sr 7840	. . . . 5 ⊢ (0R <R 𝑦 → ∃𝑧 ∈ R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15		opelreal 7894	. . . . . . . . . 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 4040	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
18		ltresr 7906	. . . . . . . . . . . . 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 7905	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
22	21	eqeq1d 2205	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23		df-1 7887	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2207	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2196	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 7818	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 7817	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 4272	. . . . . . . . . . . . . . 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 4037	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (0 <ℝ 𝑥 ↔ 0 <ℝ ⟨𝑧, 0R⟩))
37		oveq2 5930	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
38	37	eqeq1d 2205	. . . . . . . . . 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 2868	. . . . . . . 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 2613	. . . . 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 2795	. 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 1506   ∈ wcel 2167  ∃wrex 2476  ⟨cop 3625   class class class wbr 4033  (class class class)co 5922  Rcnr 7364  0_Rc0r 7365  1_Rc1r 7366   ·_R cmr 7369   <_R cltr 7370  ℝcr 7878  0cc0 7879  1c1 7880   <_ℝ cltrr 7883   · cmul 7884

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-iltp 7537  df-enr 7793  df-nr 7794  df-plr 7795  df-mr 7796  df-ltr 7797  df-0r 7798  df-1r 7799  df-m1r 7800  df-c 7885  df-0 7886  df-1 7887  df-r 7889  df-mul 7891  df-lt 7892

This theorem is referenced by:  rereceu  7956  recriota  7957