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Theorem axmulcl 7856
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 7900 be used later. Instead, in most cases use mulcl 7929. (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.
StepHypRef Expression
1 elxpi 4639 . . . . 5 (𝐴 ∈ (R × R) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)))
2 df-c 7808 . . . . 5 ℂ = (R × R)
31, 2eleq2s 2272 . . . 4 (𝐴 ∈ ℂ → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)))
4 elxpi 4639 . . . . 5 (𝐵 ∈ (R × R) → ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R)))
54, 2eleq2s 2272 . . . 4 (𝐵 ∈ ℂ → ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R)))
63, 5anim12i 338 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
7 ee4anv 1934 . . 3 (∃𝑥𝑦𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) ↔ (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
86, 7sylibr 134 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑥𝑦𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
9 simpll 527 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝐴 = ⟨𝑥, 𝑦⟩)
10 simprl 529 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
119, 10oveq12d 5887 . . . . . 6 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) = (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩))
12 mulcnsr 7825 . . . . . . 7 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
1312ad2ant2l 508 . . . . . 6 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
1411, 13eqtrd 2210 . . . . 5 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
15 simplrl 535 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑥R)
16 simprrl 539 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑧R)
17 mulclsr 7744 . . . . . . . . 9 ((𝑥R𝑧R) → (𝑥 ·R 𝑧) ∈ R)
1815, 16, 17syl2anc 411 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑥 ·R 𝑧) ∈ R)
19 m1r 7742 . . . . . . . . . 10 -1RR
2019a1i 9 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → -1RR)
21 simplrr 536 . . . . . . . . . 10 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑦R)
22 simprrr 540 . . . . . . . . . 10 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑤R)
23 mulclsr 7744 . . . . . . . . . 10 ((𝑦R𝑤R) → (𝑦 ·R 𝑤) ∈ R)
2421, 22, 23syl2anc 411 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑦 ·R 𝑤) ∈ R)
25 mulclsr 7744 . . . . . . . . 9 ((-1RR ∧ (𝑦 ·R 𝑤) ∈ R) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
2620, 24, 25syl2anc 411 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
27 addclsr 7743 . . . . . . . 8 (((𝑥 ·R 𝑧) ∈ R ∧ (-1R ·R (𝑦 ·R 𝑤)) ∈ R) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
2818, 26, 27syl2anc 411 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
29 mulclsr 7744 . . . . . . . . 9 ((𝑦R𝑧R) → (𝑦 ·R 𝑧) ∈ R)
3021, 16, 29syl2anc 411 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑦 ·R 𝑧) ∈ R)
31 mulclsr 7744 . . . . . . . . 9 ((𝑥R𝑤R) → (𝑥 ·R 𝑤) ∈ R)
3215, 22, 31syl2anc 411 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑥 ·R 𝑤) ∈ R)
33 addclsr 7743 . . . . . . . 8 (((𝑦 ·R 𝑧) ∈ R ∧ (𝑥 ·R 𝑤) ∈ R) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
3430, 32, 33syl2anc 411 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
35 opelxpi 4655 . . . . . . 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))
3628, 34, 35syl2anc 411 . . . . . 6 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩ ∈ (R × R))
3736, 2eleqtrrdi 2271 . . . . 5 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩ ∈ ℂ)
3814, 37eqeltrd 2254 . . . 4 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) ∈ ℂ)
3938exlimivv 1896 . . 3 (∃𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) ∈ ℂ)
4039exlimivv 1896 . 2 (∃𝑥𝑦𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) ∈ ℂ)
418, 40syl 14 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  cop 3594   × cxp 4621  (class class class)co 5869  Rcnr 7287  -1Rcm1r 7290   +R cplr 7291   ·R cmr 7292  cc 7800   · cmul 7807
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-i1p 7457  df-iplp 7458  df-imp 7459  df-enr 7716  df-nr 7717  df-plr 7718  df-mr 7719  df-m1r 7723  df-c 7808  df-mul 7814
This theorem is referenced by:  axmulf  7859
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