Theorem axprecex 7688

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 7730.

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 7636	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 2422	. . . 4 ⊢ (∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
3	1, 2	bitri 183	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4		breq2 3933	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐴))
5		oveq1 5781	. . . . . . 7 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2148	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
7	6	anbi2d 459	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)))
8	7	rexbidv 2438	. . . 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 233	. . 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 7627	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 3936	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
12		ltresr 7647	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
13	11, 12	bitri 183	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
14		recexgt0sr 7581	. . . . 5 ⊢ (0R <R 𝑦 → ∃𝑧 ∈ R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15		opelreal 7635	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
16	15	anbi1i 453	. . . . . . . . 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 3936	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
18		ltresr 7647	. . . . . . . . . . . . 13 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩ ↔ 0R <R 𝑧)
19	17, 18	bitri 183	. . . . . . . . . . . 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 7646	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
22	21	eqeq1d 2148	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23		df-1 7628	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2150	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2139	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 7559	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 7558	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 4161	. . . . . . . . . . . . . . 15 ⊢ ((1R ∈ R ∧ 0R ∈ R) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
29	26, 27, 28	mp2an 422	. . . . . . . . . . . . . 14 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
30	25, 29	mpbiran2 925	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
31	24, 30	bitri 183	. . . . . . . . . . . 12 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
32	22, 31	syl6bb 195	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
33	20, 32	anbi12d 464	. . . . . . . . . 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 447	. . . . . . . . 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 191	. . . . . . . 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 3933	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (0 <ℝ 𝑥 ↔ 0 <ℝ ⟨𝑧, 0R⟩))
37		oveq2 5782	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
38	37	eqeq1d 2148	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
39	36, 38	anbi12d 464	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <ℝ ⟨𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
40	39	rspcev 2789	. . . . . . . 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 163	. . . . . . 7 ⊢ (𝑦 ∈ R → ((𝑧 ∈ R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
42	41	expd 256	. . . . . 6 ⊢ (𝑦 ∈ R → (𝑧 ∈ R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
43	42	rexlimdv 2548	. . . . 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	syl5bi 151	. . 3 ⊢ (𝑦 ∈ R → (0 <ℝ ⟨𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
46	3, 9, 45	gencl 2718	. 2 ⊢ (𝐴 ∈ ℝ → (0 <ℝ 𝐴 → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)))
47	46	imp 123	1 ⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929  (class class class)co 5774  Rcnr 7105  0_Rc0r 7106  1_Rc1r 7107   ·_R cmr 7110   <_R cltr 7111  ℝcr 7619  0cc0 7620  1c1 7621   <_ℝ cltrr 7624   · cmul 7625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-imp 7277  df-iltp 7278  df-enr 7534  df-nr 7535  df-plr 7536  df-mr 7537  df-ltr 7538  df-0r 7539  df-1r 7540  df-m1r 7541  df-c 7626  df-0 7627  df-1 7628  df-r 7630  df-mul 7632  df-lt 7633

This theorem is referenced by:  rereceu  7697  recriota  7698