Theorem axprecex 8200

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 8242.

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 8148	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 2528	. . . 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 4115	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0 <ℝ 𝐴))
5		oveq1 6059	. . . . . . 7 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2243	. . . . . 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 2545	. . . 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 8139	. . . . . 6 ⊢ 0 = ⟨0R, 0R⟩
11	10	breq1i 4118	. . . . 5 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩)
12		ltresr 8159	. . . . 5 ⊢ (⟨0R, 0R⟩ <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
13	11, 12	bitri 184	. . . 4 ⊢ (0 <ℝ ⟨𝑦, 0R⟩ ↔ 0R <R 𝑦)
14		recexgt0sr 8093	. . . . 5 ⊢ (0R <R 𝑦 → ∃𝑧 ∈ R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15		opelreal 8147	. . . . . . . . . 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 4118	. . . . . . . . . . . . 13 ⊢ (0 <ℝ ⟨𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨𝑧, 0R⟩)
18		ltresr 8159	. . . . . . . . . . . . 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 8158	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
22	21	eqeq1d 2243	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23		df-1 8140	. . . . . . . . . . . . . 14 ⊢ 1 = ⟨1R, 0R⟩
24	23	eqeq2i 2245	. . . . . . . . . . . . 13 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25		eqid 2234	. . . . . . . . . . . . . 14 ⊢ 0R = 0R
26		1sr 8071	. . . . . . . . . . . . . . 15 ⊢ 1R ∈ R
27		0r 8070	. . . . . . . . . . . . . . 15 ⊢ 0R ∈ R
28		opthg2 4357	. . . . . . . . . . . . . . 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 950	. . . . . . . . . . . . 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 4115	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (0 <ℝ 𝑥 ↔ 0 <ℝ ⟨𝑧, 0R⟩))
37		oveq2 6060	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
38	37	eqeq1d 2243	. . . . . . . . . 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 2923	. . . . . . . 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 2661	. . . . 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 2848	. 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 1398  ∃wex 1541   ∈ wcel 2205  ∃wrex 2523  ⟨cop 3694   class class class wbr 4111  (class class class)co 6052  Rcnr 7617  0_Rc0r 7618  1_Rc1r 7619   ·_R cmr 7622   <_R cltr 7623  ℝcr 8131  0cc0 8132  1c1 8133   <_ℝ cltrr 8136   · cmul 8137

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673  df-enq0 7744  df-nq0 7745  df-0nq0 7746  df-plq0 7747  df-mq0 7748  df-inp 7786  df-i1p 7787  df-iplp 7788  df-imp 7789  df-iltp 7790  df-enr 8046  df-nr 8047  df-plr 8048  df-mr 8049  df-ltr 8050  df-0r 8051  df-1r 8052  df-m1r 8053  df-c 8138  df-0 8139  df-1 8140  df-r 8142  df-mul 8144  df-lt 8145

This theorem is referenced by:  rereceu  8209  recriota  8210