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Theorem axmulcl 7382
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 7422 be used later. Instead, in most cases use mulcl 7448. (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 4444 . . . . 5 (𝐴 ∈ (R × R) → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)))
2 df-c 7335 . . . . 5 ℂ = (R × R)
31, 2eleq2s 2182 . . . 4 (𝐴 ∈ ℂ → ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)))
4 elxpi 4444 . . . . 5 (𝐵 ∈ (R × R) → ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R)))
54, 2eleq2s 2182 . . . 4 (𝐵 ∈ ℂ → ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R)))
63, 5anim12i 331 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
7 ee4anv 1857 . . 3 (∃𝑥𝑦𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) ↔ (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ ∃𝑧𝑤(𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
86, 7sylibr 132 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ∃𝑥𝑦𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))))
9 simpll 496 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝐴 = ⟨𝑥, 𝑦⟩)
10 simprl 498 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝐵 = ⟨𝑧, 𝑤⟩)
119, 10oveq12d 5652 . . . . . 6 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) = (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩))
12 mulcnsr 7351 . . . . . . 7 (((𝑥R𝑦R) ∧ (𝑧R𝑤R)) → (⟨𝑥, 𝑦⟩ · ⟨𝑧, 𝑤⟩) = ⟨((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))), ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤))⟩)
1312ad2ant2l 492 . . . . . 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 2120 . . . . 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 502 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑥R)
16 simprrl 506 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑧R)
17 mulclsr 7279 . . . . . . . . 9 ((𝑥R𝑧R) → (𝑥 ·R 𝑧) ∈ R)
1815, 16, 17syl2anc 403 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑥 ·R 𝑧) ∈ R)
19 m1r 7277 . . . . . . . . . 10 -1RR
2019a1i 9 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → -1RR)
21 simplrr 503 . . . . . . . . . 10 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑦R)
22 simprrr 507 . . . . . . . . . 10 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → 𝑤R)
23 mulclsr 7279 . . . . . . . . . 10 ((𝑦R𝑤R) → (𝑦 ·R 𝑤) ∈ R)
2421, 22, 23syl2anc 403 . . . . . . . . 9 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑦 ·R 𝑤) ∈ R)
25 mulclsr 7279 . . . . . . . . 9 ((-1RR ∧ (𝑦 ·R 𝑤) ∈ R) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
2620, 24, 25syl2anc 403 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (-1R ·R (𝑦 ·R 𝑤)) ∈ R)
27 addclsr 7278 . . . . . . . 8 (((𝑥 ·R 𝑧) ∈ R ∧ (-1R ·R (𝑦 ·R 𝑤)) ∈ R) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
2818, 26, 27syl2anc 403 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ((𝑥 ·R 𝑧) +R (-1R ·R (𝑦 ·R 𝑤))) ∈ R)
29 mulclsr 7279 . . . . . . . . 9 ((𝑦R𝑧R) → (𝑦 ·R 𝑧) ∈ R)
3021, 16, 29syl2anc 403 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑦 ·R 𝑧) ∈ R)
31 mulclsr 7279 . . . . . . . . 9 ((𝑥R𝑤R) → (𝑥 ·R 𝑤) ∈ R)
3215, 22, 31syl2anc 403 . . . . . . . 8 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝑥 ·R 𝑤) ∈ R)
33 addclsr 7278 . . . . . . . 8 (((𝑦 ·R 𝑧) ∈ R ∧ (𝑥 ·R 𝑤) ∈ R) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
3430, 32, 33syl2anc 403 . . . . . . 7 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → ((𝑦 ·R 𝑧) +R (𝑥 ·R 𝑤)) ∈ R)
35 opelxpi 4459 . . . . . . 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 403 . . . . . 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, 2syl6eleqr 2181 . . . . 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 2164 . . . 4 (((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) ∈ ℂ)
3938exlimivv 1824 . . 3 (∃𝑧𝑤((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥R𝑦R)) ∧ (𝐵 = ⟨𝑧, 𝑤⟩ ∧ (𝑧R𝑤R))) → (𝐴 · 𝐵) ∈ ℂ)
4039exlimivv 1824 . 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 102   = wceq 1289  wex 1426  wcel 1438  cop 3444   × cxp 4426  (class class class)co 5634  Rcnr 6835  -1Rcm1r 6838   +R cplr 6839   ·R cmr 6840  cc 7327   · cmul 7334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-i1p 7005  df-iplp 7006  df-imp 7007  df-enr 7251  df-nr 7252  df-plr 7253  df-mr 7254  df-m1r 7258  df-c 7335  df-mul 7341
This theorem is referenced by: (None)
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