Theorem axmulcl 8186

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 8230 be used later. Instead, in most cases use mulcl 8259. (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 4767	. . . . 5 ⊢ (𝐴 ∈ (R × R) → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
2		df-c 8138	. . . . 5 ⊢ ℂ = (R × R)
3	1, 2	eleq2s 2329	. . . 4 ⊢ (𝐴 ∈ ℂ → ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)))
4		elxpi 4767	. . . . 5 ⊢ (𝐵 ∈ (R × R) → ∃𝑧∃𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R)))
5	4, 2	eleq2s 2329	. . . 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 1990	. . 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 527	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐴 = ⟨𝑥, 𝑦⟩)
10		simprl 531	. . . . . . 7 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
11	9, 10	oveq12d 6070	. . . . . 6 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) = (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩))
12		mulcnsr 8155	. . . . . . 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 508	. . . . . 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 2267	. . . . 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 537	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑥 ∈ R)
16		simprrl 541	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑧 ∈ R)
17		mulclsr 8074	. . . . . . . . 9 ⊢ ((𝑥 ∈ R ∧ 𝑧 ∈ R) → (𝑥 ·R 𝑧) ∈ R)
18	15, 16, 17	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑥 ·R 𝑧) ∈ R)
19		m1r 8072	. . . . . . . . . 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 538	. . . . . . . . . 10 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑦 ∈ R)
22		simprrr 542	. . . . . . . . . 10 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → 𝑤 ∈ R)
23		mulclsr 8074	. . . . . . . . . 10 ⊢ ((𝑦 ∈ R ∧ 𝑤 ∈ R) → (𝑦 ·R 𝑤) ∈ R)
24	21, 22, 23	syl2anc 411	. . . . . . . . 9 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑦 ·R 𝑤) ∈ R)
25		mulclsr 8074	. . . . . . . . 9 ⊢ ((-1R ∈ R ∧ (𝑦 ·R 𝑤) ∈ R) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
26	20, 24, 25	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
27		addclsr 8073	. . . . . . . 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 411	. . . . . . 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 8074	. . . . . . . . 9 ⊢ ((𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑦 ·R 𝑧) ∈ R)
30	21, 16, 29	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑦 ·R 𝑧) ∈ R)
31		mulclsr 8074	. . . . . . . . 9 ⊢ ((𝑥 ∈ R ∧ 𝑤 ∈ R) → (𝑥 ·R 𝑤) ∈ R)
32	15, 22, 31	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝑥 ·R 𝑤) ∈ R)
33		addclsr 8073	. . . . . . . 8 ⊢ (((𝑦 ·R 𝑧) ∈ R ∧ (𝑥 ·R 𝑤) ∈ R) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
34	30, 32, 33	syl2anc 411	. . . . . . 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 4783	. . . . . . 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 411	. . . . . 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 2328	. . . . 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 2311	. . . 4 ⊢ (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) ∈ ℂ)
39	38	exlimivv 1948	. . 3 ⊢ (∃𝑧∃𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ R ∧ 𝑤 ∈ R))) → (𝐴 · 𝐵) ∈ ℂ)
40	39	exlimivv 1948	. 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 1398  ∃wex 1541   ∈ wcel 2205  ⟨cop 3694   × cxp 4749  (class class class)co 6052  Rcnr 7617  -1_Rcm1r 7620   +_R cplr 7621   ·_R cmr 7622  ℂcc 8130   · cmul 8137

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673  df-enq0 7744  df-nq0 7745  df-0nq0 7746  df-plq0 7747  df-mq0 7748  df-inp 7786  df-i1p 7787  df-iplp 7788  df-imp 7789  df-enr 8046  df-nr 8047  df-plr 8048  df-mr 8049  df-m1r 8053  df-c 8138  df-mul 8144

This theorem is referenced by:  axmulf  8189