Theorem axmulcl 8227

Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 8271 be used later. Instead, in most cases use mulcl 8300. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
axmulcl	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Proof of Theorem axmulcl

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxpi 4788	. . . . 5 ⊢ (𝐴 ∈ (R × R) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
2		df-c 8179	. . . . 5 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2333	. . . 4 ⊢ (𝐴 ∈ ℂ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
4		elxpi 4788	. . . . 5 ⊢ (𝐵 ∈ (R × R) → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
5	4, 2	eleq2s 2333	. . . 4 ⊢ (𝐵 ∈ ℂ → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
6	3, 5	anim12i 338	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
7		ee4anv 1994	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) ↔ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
8	6, 7	sylibr 134	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))))
9		simpll 531	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐴 = ⟨𝑥, 𝑦⟩)
10		simprl 535	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
11	9, 10	oveq12d 6097	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) = (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩))
12		mulcnsr 8196	. . . . . . 7 ⊢ (((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)) → (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
13	12	ad2ant2l 512	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
14	11, 13	eqtrd 2271	. . . . 5 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
15		simplrl 541	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑥 ∈ R)
16		simprrl 545	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑧 ∈ R)
17		mulclsr 8115	. . . . . . . . 9 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 ·R 𝑧) ∈ R)
18	15, 16, 17	syl2anc 415	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑥 ·R 𝑧) ∈ R)
19		m1r 8113	. . . . . . . . . 10 ⊢ -1R ∈ R
20	19	a1i 9	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → -1R ∈ R)
21		simplrr 542	. . . . . . . . . 10 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑦 ∈ R)
22		simprrr 546	. . . . . . . . . 10 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑤 ∈ R)
23		mulclsr 8115	. . . . . . . . . 10 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 ·R 𝑤) ∈ R)
24	21, 22, 23	syl2anc 415	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑦 ·R 𝑤) ∈ R)
25		mulclsr 8115	. . . . . . . . 9 ⊢ ((-1R ∈ R ∧ (𝑦 ·R 𝑤) ∈ R) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
26	20, 24, 25	syl2anc 415	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
27		addclsr 8114	. . . . . . . 8 ⊢ (((𝑥 ·R 𝑧) ∈ R ∧ (-1R ·R (𝑦 ·R 𝑤)) ∈ R) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
28	18, 26, 27	syl2anc 415	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
29		mulclsr 8115	. . . . . . . . 9 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑦 ·R 𝑧) ∈ R)
30	21, 16, 29	syl2anc 415	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑦 ·R 𝑧) ∈ R)
31		mulclsr 8115	. . . . . . . . 9 ⊢ ((𝑥 ∈ R ∧ 𝑤 ∈ R) → (𝑥 ·R 𝑤) ∈ R)
32	15, 22, 31	syl2anc 415	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑥 ·R 𝑤) ∈ R)
33		addclsr 8114	. . . . . . . 8 ⊢ (((𝑦 ·R 𝑧) ∈ R ∧ (𝑥 ·R 𝑤) ∈ R) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
34	30, 32, 33	syl2anc 415	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
35		opelxpi 4804	. . . . . . 7 ⊢ ((((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R ∧ ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R) → ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩ ∈ (R × R))
36	28, 34, 35	syl2anc 415	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩ ∈ (R × R))
37	36, 2	eleqtrrdi 2332	. . . . 5 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩ ∈ ℂ)
38	14, 37	eqeltrd 2315	. . . 4 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) ∈ ℂ)
39	38	exlimivv 1952	. . 3 ⊢ (∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) ∈ ℂ)
40	39	exlimivv 1952	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) ∈ ℂ)
41	8, 40	syl 14	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3711   × cxp 4770  (class class class)co 6079  Rcnr 7658  -1_Rcm1r 7661   +_R cplr 7662   ·_R cmr 7663  ℂcc 8171   · cmul 8178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-2o 6682  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-enq0 7785  df-nq0 7786  df-0nq0 7787  df-plq0 7788  df-mq0 7789  df-inp 7827  df-i1p 7828  df-iplp 7829  df-imp 7830  df-enr 8087  df-nr 8088  df-plr 8089  df-mr 8090  df-m1r 8094  df-c 8179  df-mul 8185

This theorem is referenced by:  axmulf  8230