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Theorem axprecex 6952
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 6992.

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 6903 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2312 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 173 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 breq2 3768 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐴))
5 oveq1 5519 . . . . . . 7 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2048 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76anbi2d 437 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
87rexbidv 2327 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
94, 8imbi12d 223 . . 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 6894 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 3771 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
12 ltresr 6913 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1311, 12bitri 173 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
14 recexgt0sr 6856 . . . . 5 (0R <R 𝑦 → ∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15 opelreal 6902 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1615anbi1i 431 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
1710breq1i 3771 . . . . . . . . . . . . 13 (0 <𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑧, 0R⟩)
18 ltresr 6913 . . . . . . . . . . . . 13 (⟨0R, 0R⟩ <𝑧, 0R⟩ ↔ 0R <R 𝑧)
1917, 18bitri 173 . . . . . . . . . . . 12 (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧)
2019a1i 9 . . . . . . . . . . 11 ((𝑦R𝑧R) → (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧))
21 mulresr 6912 . . . . . . . . . . . . 13 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2221eqeq1d 2048 . . . . . . . . . . . 12 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23 df-1 6895 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2050 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2040 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 6834 . . . . . . . . . . . . . . 15 1RR
27 0r 6833 . . . . . . . . . . . . . . 15 0RR
28 opthg2 3976 . . . . . . . . . . . . . . 15 ((1RR ∧ 0RR) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
2926, 27, 28mp2an 402 . . . . . . . . . . . . . 14 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
3025, 29mpbiran2 848 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
3124, 30bitri 173 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
3222, 31syl6bb 185 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
3320, 32anbi12d 442 . . . . . . . . . 10 ((𝑦R𝑧R) → ((0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)))
3433pm5.32da 425 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
3516, 34syl5bb 181 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
36 breq2 3768 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (0 < 𝑥 ↔ 0 <𝑧, 0R⟩))
37 oveq2 5520 . . . . . . . . . . 11 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3837eqeq1d 2048 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3936, 38anbi12d 442 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
4039rspcev 2656 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))
4135, 40syl6bir 153 . . . . . . 7 (𝑦R → ((𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4241expd 245 . . . . . 6 (𝑦R → (𝑧R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
4342rexlimdv 2432 . . . . 5 (𝑦R → (∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4414, 43syl5 28 . . . 4 (𝑦R → (0R <R 𝑦 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4513, 44syl5bi 141 . . 3 (𝑦R → (0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
463, 9, 45gencl 2586 . 2 (𝐴 ∈ ℝ → (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
4746imp 115 1 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  wrex 2307  cop 3378   class class class wbr 3764  (class class class)co 5512  Rcnr 6393  0Rc0r 6394  1Rc1r 6395   ·R cmr 6398   <R cltr 6399  cr 6886  0cc0 6887  1c1 6888   < cltrr 6891   · cmul 6892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-imp 6565  df-iltp 6566  df-enr 6809  df-nr 6810  df-plr 6811  df-mr 6812  df-ltr 6813  df-0r 6814  df-1r 6815  df-m1r 6816  df-c 6893  df-0 6894  df-1 6895  df-r 6897  df-mul 6899  df-lt 6900
This theorem is referenced by:  rereceu  6961  recriota  6962
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