Theorem axrrecex 11086

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 11110. (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 11054	. . . 4 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑦 ∈ R ⟨𝑦, 0R⟩ = 𝐴)
2		df-rex 3063	. . . 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 2995	. . . 4 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝐴 ≠ 0))
5		oveq1 7375	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → (⟨𝑦, 0R⟩ · 𝑥) = (𝐴 · 𝑥))
6	5	eqeq1d 2739	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 𝐴 → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (𝐴 · 𝑥) = 1))
7	6	rexbidv 3162	. . . 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 11045	. . . . . . 7 ⊢ 0 = ⟨0R, 0R⟩
10	9	eqeq2i 2750	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = 0 ↔ ⟨𝑦, 0R⟩ = ⟨0R, 0R⟩)
11		vex 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
12	11	eqresr 11060	. . . . . 6 ⊢ (⟨𝑦, 0R⟩ = ⟨0R, 0R⟩ ↔ 𝑦 = 0R)
13	10, 12	bitri 275	. . . . 5 ⊢ (⟨𝑦, 0R⟩ = 0 ↔ 𝑦 = 0R)
14	13	necon3bii 2985	. . . 4 ⊢ (⟨𝑦, 0R⟩ ≠ 0 ↔ 𝑦 ≠ 0R)
15		recexsr 11030	. . . . . 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 11053	. . . . . . . . . 10 ⊢ (⟨𝑧, 0R⟩ ∈ ℝ ↔ 𝑧 ∈ R)
18	17	anbi1i 625	. . . . . . . . 9 ⊢ ((⟨𝑧, 0R⟩ ∈ ℝ ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1) ↔ (𝑧 ∈ R ∧ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
19		mulresr 11062	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = ⟨(𝑦 ·R 𝑧), 0R⟩)
20	19	eqeq1d 2739	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → ((⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = 1))
21		df-1 11046	. . . . . . . . . . . . 13 ⊢ 1 = ⟨1R, 0R⟩
22	21	eqeq2i 2750	. . . . . . . . . . . 12 ⊢ (⟨(𝑦 ·R 𝑧), 0R⟩ = 1 ↔ ⟨(𝑦 ·R 𝑧), 0R⟩ = ⟨1R, 0R⟩)
23		ovex 7401	. . . . . . . . . . . . 13 ⊢ (𝑦 ·R 𝑧) ∈ V
24	23	eqresr 11060	. . . . . . . . . . . 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 7376	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → (⟨𝑦, 0R⟩ · 𝑥) = (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩))
30	29	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑧, 0R⟩ → ((⟨𝑦, 0R⟩ · 𝑥) = 1 ↔ (⟨𝑦, 0R⟩ · ⟨𝑧, 0R⟩) = 1))
31	30	rspcev 3578	. . . . . . . 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 3137	. . . . 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 3484	. 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 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  ⟨cop 4588  (class class class)co 7368  Rcnr 10788  0_Rc0r 10789  1_Rc1r 10790   ·_R cmr 10793  ℝcr 11037  0cc0 11038  1c1 11039   · cmul 11043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ec 8647  df-qs 8651  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-1p 10905  df-plp 10906  df-mp 10907  df-ltp 10908  df-enr 10978  df-nr 10979  df-plr 10980  df-mr 10981  df-ltr 10982  df-0r 10983  df-1r 10984  df-m1r 10985  df-c 11044  df-0 11045  df-1 11046  df-r 11048  df-mul 11050

This theorem is referenced by: (None)