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Theorem axprecex 7512
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 7552.

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 7463 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2376 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 183 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 breq2 3871 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (0 <𝑦, 0R⟩ ↔ 0 < 𝐴))
5 oveq1 5697 . . . . . . 7 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2103 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76anbi2d 453 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
87rexbidv 2392 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
94, 8imbi12d 233 . . 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 7454 . . . . . 6 0 = ⟨0R, 0R
1110breq1i 3874 . . . . 5 (0 <𝑦, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑦, 0R⟩)
12 ltresr 7473 . . . . 5 (⟨0R, 0R⟩ <𝑦, 0R⟩ ↔ 0R <R 𝑦)
1311, 12bitri 183 . . . 4 (0 <𝑦, 0R⟩ ↔ 0R <R 𝑦)
14 recexgt0sr 7416 . . . . 5 (0R <R 𝑦 → ∃𝑧R (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))
15 opelreal 7462 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1615anbi1i 447 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
1710breq1i 3874 . . . . . . . . . . . . 13 (0 <𝑧, 0R⟩ ↔ ⟨0R, 0R⟩ <𝑧, 0R⟩)
18 ltresr 7473 . . . . . . . . . . . . 13 (⟨0R, 0R⟩ <𝑧, 0R⟩ ↔ 0R <R 𝑧)
1917, 18bitri 183 . . . . . . . . . . . 12 (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧)
2019a1i 9 . . . . . . . . . . 11 ((𝑦R𝑧R) → (0 <𝑧, 0R⟩ ↔ 0R <R 𝑧))
21 mulresr 7472 . . . . . . . . . . . . 13 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2221eqeq1d 2103 . . . . . . . . . . . 12 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
23 df-1 7455 . . . . . . . . . . . . . 14 1 = ⟨1R, 0R
2423eqeq2i 2105 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
25 eqid 2095 . . . . . . . . . . . . . 14 0R = 0R
26 1sr 7394 . . . . . . . . . . . . . . 15 1RR
27 0r 7393 . . . . . . . . . . . . . . 15 0RR
28 opthg2 4090 . . . . . . . . . . . . . . 15 ((1RR ∧ 0RR) → (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R)))
2926, 27, 28mp2an 418 . . . . . . . . . . . . . 14 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ ((𝑦 ·R 𝑧) = 1R ∧ 0R = 0R))
3025, 29mpbiran2 890 . . . . . . . . . . . . 13 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
3124, 30bitri 183 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
3222, 31syl6bb 195 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
3320, 32anbi12d 458 . . . . . . . . . 10 ((𝑦R𝑧R) → ((0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)))
3433pm5.32da 441 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
3516, 34syl5bb 191 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) ↔ (𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R))))
36 breq2 3871 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (0 < 𝑥 ↔ 0 <𝑧, 0R⟩))
37 oveq2 5698 . . . . . . . . . . 11 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3837eqeq1d 2103 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3936, 38anbi12d 458 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)))
4039rspcev 2736 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (0 <𝑧, 0R⟩ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))
4135, 40syl6bir 163 . . . . . . 7 (𝑦R → ((𝑧R ∧ (0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R)) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
4241expd 255 . . . . . 6 (𝑦R → (𝑧R → ((0R <R 𝑧 ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1))))
4342rexlimdv 2501 . . . . 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, 44syl5bi 151 . . 3 (𝑦R → (0 <𝑦, 0R⟩ → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
463, 9, 45gencl 2665 . 2 (𝐴 ∈ ℝ → (0 < 𝐴 → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1)))
4746imp 123 1 ((𝐴 ∈ ℝ ∧ 0 < 𝐴) → ∃𝑥 ∈ ℝ (0 < 𝑥 ∧ (𝐴 · 𝑥) = 1))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wex 1433  wcel 1445  wrex 2371  cop 3469   class class class wbr 3867  (class class class)co 5690  Rcnr 6953  0Rc0r 6954  1Rc1r 6955   ·R cmr 6958   <R cltr 6959  cr 7446  0cc0 7447  1c1 7448   < cltrr 7451   · cmul 7452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431
This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-2o 6220  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-enq0 7080  df-nq0 7081  df-0nq0 7082  df-plq0 7083  df-mq0 7084  df-inp 7122  df-i1p 7123  df-iplp 7124  df-imp 7125  df-iltp 7126  df-enr 7369  df-nr 7370  df-plr 7371  df-mr 7372  df-ltr 7373  df-0r 7374  df-1r 7375  df-m1r 7376  df-c 7453  df-0 7454  df-1 7455  df-r 7457  df-mul 7459  df-lt 7460
This theorem is referenced by:  rereceu  7521  recriota  7522
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