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Theorem axprecex 7909
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 7951.

In treatments which assume excluded middle, the 0 < 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 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 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.
StepHypRef Expression
1 elreal 7857 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2474 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 184 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 breq2 4022 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐴))
5 oveq1 5903 . . . . . . 7 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2198 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76anbi2d 464 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
87rexbidv 2491 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
94, 8imbi12d 234 . . 3 (⟨𝑦, 0R⟩ = 𝐴 → ((0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)) ↔ (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))))
10 df-0 7848 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 4025 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
12 ltresr 7868 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1311, 12bitri 184 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
14 recexgt0sr 7802 . . . . 5 (0R <R 𝑦 → ∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15 opelreal 7856 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1615anbi1i 458 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
1710breq1i 4025 . . . . . . . . . . . . 13 (0 <𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑧, 0R⟩)
18 ltresr 7868 . . . . . . . . . . . . 13 (⟨0R, 0R⟩ <𝑧, 0R⟩ ↔ 0R <R 𝑧)
1917, 18bitri 184 . . . . . . . . . . . 12 (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧)
2019a1i 9 . . . . . . . . . . 11 ((𝑦R𝑧R) → (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧))
21 mulresr 7867 . . . . . . . . . . . . 13 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2221eqeq1d 2198 . . . . . . . . . . . 12 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23 df-1 7849 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2200 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2189 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 7780 . . . . . . . . . . . . . . 15 1RR
27 0r 7779 . . . . . . . . . . . . . . 15 0RR
28 opthg2 4257 . . . . . . . . . . . . . . 15 ((1RR ∧ 0RR) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
2926, 27, 28mp2an 426 . . . . . . . . . . . . . 14 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
3025, 29mpbiran2 943 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
3124, 30bitri 184 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
3222, 31bitrdi 196 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
3320, 32anbi12d 473 . . . . . . . . . 10 ((𝑦R𝑧R) → ((0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)))
3433pm5.32da 452 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
3516, 34bitrid 192 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
36 breq2 4022 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (0 < 𝑥 ↔ 0 <𝑧, 0R⟩))
37 oveq2 5904 . . . . . . . . . . 11 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3837eqeq1d 2198 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3936, 38anbi12d 473 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
4039rspcev 2856 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))
4135, 40biimtrrdi 164 . . . . . . 7 (𝑦R → ((𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4241expd 258 . . . . . 6 (𝑦R → (𝑧R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
4342rexlimdv 2606 . . . . 5 (𝑦R → (∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4414, 43syl5 32 . . . 4 (𝑦R → (0R <R 𝑦 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4513, 44biimtrid 152 . . 3 (𝑦R → (0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
463, 9, 45gencl 2784 . 2 (𝐴 ∈ ℝ → (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
4746imp 124 1 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  wrex 2469  cop 3610   class class class wbr 4018  (class class class)co 5896  Rcnr 7326  0Rc0r 7327  1Rc1r 7328   ·R cmr 7331   <R cltr 7332  cr 7840  0cc0 7841  1c1 7842   < cltrr 7845   · cmul 7846
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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-recs 6330  df-irdg 6395  df-1o 6441  df-2o 6442  df-oadd 6445  df-omul 6446  df-er 6559  df-ec 6561  df-qs 6565  df-ni 7333  df-pli 7334  df-mi 7335  df-lti 7336  df-plpq 7373  df-mpq 7374  df-enq 7376  df-nqqs 7377  df-plqqs 7378  df-mqqs 7379  df-1nqqs 7380  df-rq 7381  df-ltnqqs 7382  df-enq0 7453  df-nq0 7454  df-0nq0 7455  df-plq0 7456  df-mq0 7457  df-inp 7495  df-i1p 7496  df-iplp 7497  df-imp 7498  df-iltp 7499  df-enr 7755  df-nr 7756  df-plr 7757  df-mr 7758  df-ltr 7759  df-0r 7760  df-1r 7761  df-m1r 7762  df-c 7847  df-0 7848  df-1 7849  df-r 7851  df-mul 7853  df-lt 7854
This theorem is referenced by:  rereceu  7918  recriota  7919
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