Theorem axrrecex 11057

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 11081. (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.

Step	Hyp	Ref	Expression
1		elreal 11025	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 3054	. . . 4 ⊢ (∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴 ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
3	1, 2	bitri 275	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦(𝑦 ∈ R ∧ ⟨𝑦, 0R⟩ = 𝐴))
4		neeq1 2987	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝐴 ≠ 0))
5		oveq1 7356	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2731	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
7	6	rexbidv 3153	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
8	4, 7	imbi12d 344	. . 3 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ ≠ 0 → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1) ↔ (𝐴 ≠ 0 → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)))
9		df-0 11016	. . . . . . 7 ⊢ 0 = ⟨0R, 0R⟩
10	9	eqeq2i 2742	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 0 ↔ ⟨𝑦, 0R⟩ = ⟨0R, 0R⟩)
11		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
12	11	eqresr 11031	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = ⟨0R, 0R⟩ ↔ 𝑦 = 0R)
13	10, 12	bitri 275	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 0 ↔ 𝑦 = 0R)
14	13	necon3bii 2977	. . . 4 ⊢ (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝑦 ≠ 0R)
15		recexsr 11001	. . . . . 6 ⊢ ((𝑦 ∈ R ∧ 𝑦 ≠ 0R) → ∃𝑧 ∈ R (𝑦 ·R 𝑧) = 1R)
16	15	ex 412	. . . . 5 ⊢ (𝑦 ∈ R → (𝑦 ≠ 0R → ∃𝑧 ∈ R (𝑦 ·R 𝑧) = 1R))
17		opelreal 11024	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
18	17	anbi1i 624	. . . . . . . . 9 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧 ∈ R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
19		mulresr 11033	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
20	19	eqeq1d 2731	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
21		df-1 11017	. . . . . . . . . . . . 13 ⊢ 1 = ⟨1R, 0R⟩
22	21	eqeq2i 2742	. . . . . . . . . . . 12 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
23		ovex 7382	. . . . . . . . . . . . 13 ⊢ (𝑦 ·R 𝑧) ∈ V
24	23	eqresr 11031	. . . . . . . . . . . 12 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩ ↔ (𝑦 ·R 𝑧) = 1R)
25	22, 24	bitri 275	. . . . . . . . . . 11 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ (𝑦 ·R 𝑧) = 1R)
26	20, 25	bitrdi 287	. . . . . . . . . 10 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ (𝑦 ·R 𝑧) = 1R))
27	26	pm5.32da 579	. . . . . . . . 9 ⊢ (𝑦 ∈ R → ((𝑧 ∈ R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧 ∈ R ∧ (𝑦 ·R 𝑧) = 1R)))
28	18, 27	bitrid 283	. . . . . . . 8 ⊢ (𝑦 ∈ R → ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧 ∈ R ∧ (𝑦 ·R 𝑧) = 1R)))
29		oveq2 7357	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
30	29	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
31	30	rspcev 3577	. . . . . . . 8 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)
32	28, 31	biimtrrdi 254	. . . . . . 7 ⊢ (𝑦 ∈ R → ((𝑧 ∈ R ∧ (𝑦 ·R 𝑧) = 1R) → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
33	32	expd 415	. . . . . 6 ⊢ (𝑦 ∈ R → (𝑧 ∈ R → ((𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1)))
34	33	rexlimdv 3128	. . . . 5 ⊢ (𝑦 ∈ R → (∃𝑧 ∈ R (𝑦 ·R 𝑧) = 1R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
35	16, 34	syld 47	. . . 4 ⊢ (𝑦 ∈ R → (𝑦 ≠ 0R → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
36	14, 35	biimtrid 242	. . 3 ⊢ (𝑦 ∈ R → (⟨𝑦, 0R⟩ ≠ 0 → ∃𝑥 ∈ ℝ (⟨𝑦, 0R⟩ · 𝑥) = 1))
37	3, 8, 36	gencl 3478	. 2 ⊢ (𝐴 ∈ ℝ → (𝐴 ≠ 0 → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1))
38	37	imp 406	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  ⟨cop 4583  (class class class)co 7349  Rcnr 10759  0_Rc0r 10760  1_Rc1r 10761   ·_R cmr 10764  ℝcr 11008  0cc0 11009  1c1 11010   · cmul 11014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-inf2 9537

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-oadd 8392  df-omul 8393  df-er 8625  df-ec 8627  df-qs 8631  df-ni 10766  df-pli 10767  df-mi 10768  df-lti 10769  df-plpq 10802  df-mpq 10803  df-ltpq 10804  df-enq 10805  df-nq 10806  df-erq 10807  df-plq 10808  df-mq 10809  df-1nq 10810  df-rq 10811  df-ltnq 10812  df-np 10875  df-1p 10876  df-plp 10877  df-mp 10878  df-ltp 10879  df-enr 10949  df-nr 10950  df-plr 10951  df-mr 10952  df-ltr 10953  df-0r 10954  df-1r 10955  df-m1r 10956  df-c 11015  df-0 11016  df-1 11017  df-r 11019  df-mul 11021

This theorem is referenced by: (None)