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Theorem axrrecex 10022
Description: Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 10046. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axrrecex ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Distinct variable group:   𝑥,𝐴

Proof of Theorem axrrecex
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 9990 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2947 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 264 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 neeq1 2885 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝐴 ≠ 0))
5 oveq1 6697 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2653 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76rexbidv 3081 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
84, 7imbi12d 333 . . 3 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ ≠ 0 → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (𝐴 ≠ 0 → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)))
9 df-0 9981 . . . . . . 7 0 = ⟨0R, 0R
109eqeq2i 2663 . . . . . 6 (⟨𝑦, 0R⟩ = 0 ↔ ⟨𝑦, 0R⟩ = ⟨0R, 0R⟩)
11 vex 3234 . . . . . . 7 𝑦 ∈ V
1211eqresr 9996 . . . . . 6 (⟨𝑦, 0R⟩ = ⟨0R, 0R⟩ ↔ 𝑦 = 0R)
1310, 12bitri 264 . . . . 5 (⟨𝑦, 0R⟩ = 0 ↔ 𝑦 = 0R)
1413necon3bii 2875 . . . 4 (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝑦 ≠ 0R)
15 recexsr 9966 . . . . . 6 ((𝑦R𝑦 ≠ 0R) → ∃𝑧R (𝑦 ·R 𝑧) = 1R)
1615ex 449 . . . . 5 (𝑦R → (𝑦 ≠ 0R → ∃𝑧R (𝑦 ·R 𝑧) = 1R))
17 opelreal 9989 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1817anbi1i 731 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
19 mulresr 9998 . . . . . . . . . . . 12 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2019eqeq1d 2653 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
21 df-1 9982 . . . . . . . . . . . . 13 1 = ⟨1R, 0R
2221eqeq2i 2663 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
23 ovex 6718 . . . . . . . . . . . . 13 (𝑦 ·R 𝑧) ∈ V
2423eqresr 9996 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
2522, 24bitri 264 . . . . . . . . . . 11 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
2620, 25syl6bb 276 . . . . . . . . . 10 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
2726pm5.32da 674 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (𝑦 ·R 𝑧) = 1R)))
2818, 27syl5bb 272 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (𝑦 ·R 𝑧) = 1R)))
29 oveq2 6698 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3029eqeq1d 2653 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3130rspcev 3340 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)
3228, 31syl6bir 244 . . . . . . 7 (𝑦R → ((𝑧R ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
3332expd 451 . . . . . 6 (𝑦R → (𝑧R → ((𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
3433rexlimdv 3059 . . . . 5 (𝑦R → (∃𝑧R (𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
3516, 34syld 47 . . . 4 (𝑦R → (𝑦 ≠ 0R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
3614, 35syl5bi 232 . . 3 (𝑦R → (⟨𝑦, 0R⟩ ≠ 0 → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
373, 8, 36gencl 3266 . 2 (𝐴 ∈ ℝ → (𝐴 ≠ 0 → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
3837imp 444 1 ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wrex 2942  cop 4216  (class class class)co 6690  Rcnr 9725  0Rc0r 9726  1Rc1r 9727   ·R cmr 9730  cr 9973  0cc0 9974  1c1 9975   · cmul 9979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-plp 9843  df-mp 9844  df-ltp 9845  df-enr 9915  df-nr 9916  df-plr 9917  df-mr 9918  df-ltr 9919  df-0r 9920  df-1r 9921  df-m1r 9922  df-c 9980  df-0 9981  df-1 9982  df-r 9984  df-mul 9986
This theorem is referenced by: (None)
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