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Theorem axmulf 10169
Description: Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 10176. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 10218. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
Assertion
Ref Expression
axmulf · :(ℂ × ℂ)⟶ℂ

Proof of Theorem axmulf
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moeq 3534 . . . . . . . . 9 ∃*𝑧 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩
21mosubop 5104 . . . . . . . 8 ∃*𝑧𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
32mosubop 5104 . . . . . . 7 ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))
4 anass 459 . . . . . . . . . . 11 (((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
542exbii 1925 . . . . . . . . . 10 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩) ↔ ∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
6 19.42vv 2035 . . . . . . . . . 10 (∃𝑢𝑓(𝑥 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
75, 6bitri 264 . . . . . . . . 9 (∃𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩) ↔ (𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
872exbii 1925 . . . . . . . 8 (∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩) ↔ ∃𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
98mobii 2641 . . . . . . 7 (∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩) ↔ ∃*𝑧𝑤𝑣(𝑥 = ⟨𝑤, 𝑣⟩ ∧ ∃𝑢𝑓(𝑦 = ⟨𝑢, 𝑓⟩ ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)))
103, 9mpbir 221 . . . . . 6 ∃*𝑧𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩)
1110moani 2674 . . . . 5 ∃*𝑧((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))
1211funoprab 6907 . . . 4 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
13 df-mul 10150 . . . . 5 · = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
1413funeqi 6052 . . . 4 (Fun · ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))})
1512, 14mpbir 221 . . 3 Fun ·
1613dmeqi 5463 . . . . 5 dom · = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))}
17 dmoprabss 6889 . . . . 5 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤𝑣𝑢𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = ⟨((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))⟩))} ⊆ (ℂ × ℂ)
1816, 17eqsstri 3784 . . . 4 dom · ⊆ (ℂ × ℂ)
19 0ncn 10156 . . . . 5 ¬ ∅ ∈ ℂ
20 df-c 10144 . . . . . . 7 ℂ = (R × R)
21 oveq1 6800 . . . . . . . 8 (⟨𝑧, 𝑤⟩ = 𝑥 → (⟨𝑧, 𝑤⟩ · ⟨𝑣, 𝑢⟩) = (𝑥 · ⟨𝑣, 𝑢⟩))
2221eleq1d 2835 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝑥 → ((⟨𝑧, 𝑤⟩ · ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 · ⟨𝑣, 𝑢⟩) ∈ (R × R)))
23 oveq2 6801 . . . . . . . 8 (⟨𝑣, 𝑢⟩ = 𝑦 → (𝑥 · ⟨𝑣, 𝑢⟩) = (𝑥 · 𝑦))
2423eleq1d 2835 . . . . . . 7 (⟨𝑣, 𝑢⟩ = 𝑦 → ((𝑥 · ⟨𝑣, 𝑢⟩) ∈ (R × R) ↔ (𝑥 · 𝑦) ∈ (R × R)))
25 mulcnsr 10159 . . . . . . . 8 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ · ⟨𝑣, 𝑢⟩) = ⟨((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))), ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢))⟩)
26 mulclsr 10107 . . . . . . . . . . 11 ((𝑧R𝑣R) → (𝑧 ·R 𝑣) ∈ R)
27 m1r 10105 . . . . . . . . . . . 12 -1RR
28 mulclsr 10107 . . . . . . . . . . . 12 ((𝑤R𝑢R) → (𝑤 ·R 𝑢) ∈ R)
29 mulclsr 10107 . . . . . . . . . . . 12 ((-1RR ∧ (𝑤 ·R 𝑢) ∈ R) → (-1R ·R (𝑤 ·R 𝑢)) ∈ R)
3027, 28, 29sylancr 575 . . . . . . . . . . 11 ((𝑤R𝑢R) → (-1R ·R (𝑤 ·R 𝑢)) ∈ R)
31 addclsr 10106 . . . . . . . . . . 11 (((𝑧 ·R 𝑣) ∈ R ∧ (-1R ·R (𝑤 ·R 𝑢)) ∈ R) → ((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))) ∈ R)
3226, 30, 31syl2an 583 . . . . . . . . . 10 (((𝑧R𝑣R) ∧ (𝑤R𝑢R)) → ((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))) ∈ R)
3332an4s 639 . . . . . . . . 9 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))) ∈ R)
34 mulclsr 10107 . . . . . . . . . . 11 ((𝑤R𝑣R) → (𝑤 ·R 𝑣) ∈ R)
35 mulclsr 10107 . . . . . . . . . . 11 ((𝑧R𝑢R) → (𝑧 ·R 𝑢) ∈ R)
36 addclsr 10106 . . . . . . . . . . 11 (((𝑤 ·R 𝑣) ∈ R ∧ (𝑧 ·R 𝑢) ∈ R) → ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢)) ∈ R)
3734, 35, 36syl2anr 584 . . . . . . . . . 10 (((𝑧R𝑢R) ∧ (𝑤R𝑣R)) → ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢)) ∈ R)
3837an42s 640 . . . . . . . . 9 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢)) ∈ R)
39 opelxpi 5288 . . . . . . . . 9 ((((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))) ∈ R ∧ ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢)) ∈ R) → ⟨((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))), ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢))⟩ ∈ (R × R))
4033, 38, 39syl2anc 573 . . . . . . . 8 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → ⟨((𝑧 ·R 𝑣) +R (-1R ·R (𝑤 ·R 𝑢))), ((𝑤 ·R 𝑣) +R (𝑧 ·R 𝑢))⟩ ∈ (R × R))
4125, 40eqeltrd 2850 . . . . . . 7 (((𝑧R𝑤R) ∧ (𝑣R𝑢R)) → (⟨𝑧, 𝑤⟩ · ⟨𝑣, 𝑢⟩) ∈ (R × R))
4220, 22, 24, 412optocl 5336 . . . . . 6 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ (R × R))
4342, 20syl6eleqr 2861 . . . . 5 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ)
4419, 43oprssdm 6962 . . . 4 (ℂ × ℂ) ⊆ dom ·
4518, 44eqssi 3768 . . 3 dom · = (ℂ × ℂ)
46 df-fn 6034 . . 3 ( · Fn (ℂ × ℂ) ↔ (Fun · ∧ dom · = (ℂ × ℂ)))
4715, 45, 46mpbir2an 690 . 2 · Fn (ℂ × ℂ)
4843rgen2a 3126 . 2 𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 · 𝑦) ∈ ℂ
49 ffnov 6911 . 2 ( · :(ℂ × ℂ)⟶ℂ ↔ ( · Fn (ℂ × ℂ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℂ (𝑥 · 𝑦) ∈ ℂ))
5047, 48, 49mpbir2an 690 1 · :(ℂ × ℂ)⟶ℂ
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wex 1852  wcel 2145  ∃*wmo 2619  wral 3061  cop 4322   × cxp 5247  dom cdm 5249  Fun wfun 6025   Fn wfn 6026  wf 6027  (class class class)co 6793  {coprab 6794  Rcnr 9889  -1Rcm1r 9892   +R cplr 9893   ·R cmr 9894  cc 10136   · cmul 10143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-omul 7718  df-er 7896  df-ec 7898  df-qs 7902  df-ni 9896  df-pli 9897  df-mi 9898  df-lti 9899  df-plpq 9932  df-mpq 9933  df-ltpq 9934  df-enq 9935  df-nq 9936  df-erq 9937  df-plq 9938  df-mq 9939  df-1nq 9940  df-rq 9941  df-ltnq 9942  df-np 10005  df-1p 10006  df-plp 10007  df-mp 10008  df-ltp 10009  df-enr 10079  df-nr 10080  df-plr 10081  df-mr 10082  df-m1r 10086  df-c 10144  df-mul 10150
This theorem is referenced by:  axmulcl  10176
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