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Theorem abvmul 18877
Description: An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a 𝐴 = (AbsVal‘𝑅)
abvf.b 𝐵 = (Base‘𝑅)
abvmul.t · = (.r𝑅)
Assertion
Ref Expression
abvmul ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))

Proof of Theorem abvmul
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . . . 8 𝐴 = (AbsVal‘𝑅)
21abvrcl 18869 . . . . . . 7 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . . 8 𝐵 = (Base‘𝑅)
4 eqid 2651 . . . . . . . 8 (+g𝑅) = (+g𝑅)
5 abvmul.t . . . . . . . 8 · = (.r𝑅)
6 eqid 2651 . . . . . . . 8 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 18867 . . . . . . 7 (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
82, 7syl 17 . . . . . 6 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 256 . . . . 5 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
109simprd 478 . . . 4 (𝐹𝐴 → ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
11 simpl 472 . . . . . . 7 (((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1211ralimi 2981 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1312adantl 481 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1413ralimi 2981 . . . 4 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1510, 14syl 17 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
16 oveq1 6697 . . . . . 6 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
1716fveq2d 6233 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑋 · 𝑦)))
18 fveq2 6229 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1918oveq1d 6705 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑦)))
2017, 19eqeq12d 2666 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦))))
21 oveq2 6698 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
2221fveq2d 6233 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 · 𝑦)) = (𝐹‘(𝑋 · 𝑌)))
23 fveq2 6229 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2423oveq2d 6706 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑌)))
2522, 24eqeq12d 2666 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
2620, 25rspc2v 3353 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
2715, 26syl5com 31 . 2 (𝐹𝐴 → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
28273impib 1281 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974   + caddc 9977   · cmul 9979  +∞cpnf 10109  cle 10113  [,)cico 12215  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  0gc0g 16147  Ringcrg 18593  AbsValcabv 18864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-abv 18865
This theorem is referenced by:  abv1z  18880  abvneg  18882  abvrec  18884  abvdiv  18885  abvdom  18886  abvres  18887  nmmul  22515  sranlm  22535  abvcxp  25349  qabvexp  25360  ostthlem2  25362  ostth2lem2  25368  ostth3  25372
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