Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  axrrecex Structured version   Visualization version   GIF version

Theorem axrrecex 9863
 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 9887. (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 9831 . . . 4 (𝐴 ∈ ℝ ↔ ∃𝑦R𝑦, 0R⟩ = 𝐴)
2 df-rex 2902 . . . 4 (∃𝑦R𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
31, 2bitri 263 . . 3 (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4 neeq1 2844 . . . 4 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝐴 ≠ 0))
5 oveq1 6556 . . . . . 6 (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
65eqeq1d 2612 . . . . 5 (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
76rexbidv 3034 . . . 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 9822 . . . . . . 7 0 = ⟨0R, 0R
109eqeq2i 2622 . . . . . 6 (⟨𝑦, 0R⟩ = 0 ↔ ⟨𝑦, 0R⟩ = ⟨0R, 0R⟩)
11 vex 3176 . . . . . . 7 𝑦 ∈ V
1211eqresr 9837 . . . . . 6 (⟨𝑦, 0R⟩ = ⟨0R, 0R⟩ ↔ 𝑦 = 0R)
1310, 12bitri 263 . . . . 5 (⟨𝑦, 0R⟩ = 0 ↔ 𝑦 = 0R)
1413necon3bii 2834 . . . 4 (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝑦 ≠ 0R)
15 recexsr 9807 . . . . . 6 ((𝑦R𝑦 ≠ 0R) → ∃𝑧R (𝑦 ·R 𝑧) = 1R)
1615ex 449 . . . . 5 (𝑦R → (𝑦 ≠ 0R → ∃𝑧R (𝑦 ·R 𝑧) = 1R))
17 opelreal 9830 . . . . . . . . . 10 (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧R)
1817anbi1i 727 . . . . . . . . 9 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
19 mulresr 9839 . . . . . . . . . . . 12 ((𝑦R𝑧R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
2019eqeq1d 2612 . . . . . . . . . . 11 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
21 df-1 9823 . . . . . . . . . . . . 13 1 = ⟨1R, 0R
2221eqeq2i 2622 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
23 ovex 6577 . . . . . . . . . . . . 13 (𝑦 ·R 𝑧) ∈ V
2423eqresr 9837 . . . . . . . . . . . 12 (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
2522, 24bitri 263 . . . . . . . . . . 11 (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
2620, 25syl6bb 275 . . . . . . . . . 10 ((𝑦R𝑧R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
2726pm5.32da 671 . . . . . . . . 9 (𝑦R → ((𝑧R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (𝑦 ·R 𝑧) = 1R)))
2818, 27syl5bb 271 . . . . . . . 8 (𝑦R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧R ∧ (𝑦 ·R 𝑧) = 1R)))
29 oveq2 6557 . . . . . . . . . 10 (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
3029eqeq1d 2612 . . . . . . . . 9 (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
3130rspcev 3282 . . . . . . . 8 ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)
3228, 31syl6bir 243 . . . . . . 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 3012 . . . . 5 (𝑦R → (∃𝑧R (𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
3516, 34syld 46 . . . 4 (𝑦R → (𝑦 ≠ 0R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
3614, 35syl5bi 231 . . 3 (𝑦R → (⟨𝑦, 0R⟩ ≠ 0 → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
373, 8, 36gencl 3208 . 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 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  ⟨cop 4131  (class class class)co 6549  Rcnr 9566  0Rc0r 9567  1Rc1r 9568   ·R cmr 9571  ℝcr 9814  0cc0 9815  1c1 9816   · cmul 9820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-1p 9683  df-plp 9684  df-mp 9685  df-ltp 9686  df-enr 9756  df-nr 9757  df-plr 9758  df-mr 9759  df-ltr 9760  df-0r 9761  df-1r 9762  df-m1r 9763  df-c 9821  df-0 9822  df-1 9823  df-r 9825  df-mul 9827 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator