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Theorem axrrecex 10187
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 10211. (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 10155 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 3067 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 264 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 neeq1 3005 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝐴 ≠ 0))
5 oveq1 6801 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2773 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76rexbidv 3200 . . . 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 10146 . . . . . . 7 0 = ⟨0R, 0R
109eqeq2i 2783 . . . . . 6 (⟨𝑦, 0R⟩ = 0 ↔ ⟨𝑦, 0R⟩ = ⟨0R, 0R⟩)
11 vex 3354 . . . . . . 7 𝑦 ∈ V
1211eqresr 10161 . . . . . 6 (⟨𝑦, 0R⟩ = ⟨0R, 0R⟩ ↔ 𝑦 = 0R)
1310, 12bitri 264 . . . . 5 (⟨𝑦, 0R⟩ = 0 ↔ 𝑦 = 0R)
1413necon3bii 2995 . . . 4 (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝑦 ≠ 0R)
15 recexsr 10131 . . . . . 6 ((𝑦R𝑦 ≠ 0R) → ∃𝑧R (𝑦 ·R 𝑧) = 1R)
1615ex 397 . . . . 5 (𝑦R → (𝑦 ≠ 0R → ∃𝑧R (𝑦 ·R 𝑧) = 1R))
17 opelreal 10154 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1817anbi1i 604 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
19 mulresr 10163 . . . . . . . . . . . 12 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2019eqeq1d 2773 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
21 df-1 10147 . . . . . . . . . . . . 13 1 = ⟨1R, 0R
2221eqeq2i 2783 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
23 ovex 6824 . . . . . . . . . . . . 13 (𝑦 ·R 𝑧) ∈ V
2423eqresr 10161 . . . . . . . . . . . 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 562 . . . . . . . . 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 6802 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3029eqeq1d 2773 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3130rspcev 3461 . . . . . . . 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 400 . . . . . 6 (𝑦R → (𝑧R → ((𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
3433rexlimdv 3178 . . . . 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 3387 . 2 (𝐴 ∈ ℝ → (𝐴 ≠ 0 → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
3837imp 393 1 ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wrex 3062  cop 4323  (class class class)co 6794  Rcnr 9890  0Rc0r 9891  1Rc1r 9892   ·R cmr 9895  cr 10138  0cc0 10139  1c1 10140   · cmul 10144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-omul 7719  df-er 7897  df-ec 7899  df-qs 7903  df-ni 9897  df-pli 9898  df-mi 9899  df-lti 9900  df-plpq 9933  df-mpq 9934  df-ltpq 9935  df-enq 9936  df-nq 9937  df-erq 9938  df-plq 9939  df-mq 9940  df-1nq 9941  df-rq 9942  df-ltnq 9943  df-np 10006  df-1p 10007  df-plp 10008  df-mp 10009  df-ltp 10010  df-enr 10080  df-nr 10081  df-plr 10082  df-mr 10083  df-ltr 10084  df-0r 10085  df-1r 10086  df-m1r 10087  df-c 10145  df-0 10146  df-1 10147  df-r 10149  df-mul 10151
This theorem is referenced by: (None)
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