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Theorem axprecex 7854
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 7896.

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 7802 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2459 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 184 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 breq2 4002 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐴))
5 oveq1 5872 . . . . . . 7 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2184 . . . . . 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 2476 . . . 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 7793 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 4005 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
12 ltresr 7813 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1311, 12bitri 184 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
14 recexgt0sr 7747 . . . . 5 (0R <R 𝑦 → ∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15 opelreal 7801 . . . . . . . . . 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 4005 . . . . . . . . . . . . 13 (0 <𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑧, 0R⟩)
18 ltresr 7813 . . . . . . . . . . . . 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 7812 . . . . . . . . . . . . 13 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2221eqeq1d 2184 . . . . . . . . . . . 12 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23 df-1 7794 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2186 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2175 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 7725 . . . . . . . . . . . . . . 15 1RR
27 0r 7724 . . . . . . . . . . . . . . 15 0RR
28 opthg2 4233 . . . . . . . . . . . . . . 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 941 . . . . . . . . . . . . 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 4002 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (0 < 𝑥 ↔ 0 <𝑧, 0R⟩))
37 oveq2 5873 . . . . . . . . . . 11 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3837eqeq1d 2184 . . . . . . . . . 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 2839 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))
4135, 40syl6bir 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 2591 . . . . 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 2767 . 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 1353  wex 1490  wcel 2146  wrex 2454  cop 3592   class class class wbr 3998  (class class class)co 5865  Rcnr 7271  0Rc0r 7272  1Rc1r 7273   ·R cmr 7276   <R cltr 7277  cr 7785  0cc0 7786  1c1 7787   < cltrr 7790   · cmul 7791
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-2o 6408  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-enq0 7398  df-nq0 7399  df-0nq0 7400  df-plq0 7401  df-mq0 7402  df-inp 7440  df-i1p 7441  df-iplp 7442  df-imp 7443  df-iltp 7444  df-enr 7700  df-nr 7701  df-plr 7702  df-mr 7703  df-ltr 7704  df-0r 7705  df-1r 7706  df-m1r 7707  df-c 7792  df-0 7793  df-1 7794  df-r 7796  df-mul 7798  df-lt 7799
This theorem is referenced by:  rereceu  7863  recriota  7864
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