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Theorem abvmul 18652
 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 18644 . . . . . . 7 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . . 8 𝐵 = (Base‘𝑅)
4 eqid 2610 . . . . . . . 8 (+g𝑅) = (+g𝑅)
5 abvmul.t . . . . . . . 8 · = (.r𝑅)
6 eqid 2610 . . . . . . . 8 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 18642 . . . . . . 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 255 . . . . 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 2936 . . . . . 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 2936 . . . 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 6556 . . . . . 6 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
1716fveq2d 6107 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑋 · 𝑦)))
18 fveq2 6103 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1918oveq1d 6564 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑦)))
2017, 19eqeq12d 2625 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦))))
21 oveq2 6557 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
2221fveq2d 6107 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 · 𝑦)) = (𝐹‘(𝑋 · 𝑌)))
23 fveq2 6103 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2423oveq2d 6565 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑌)))
2522, 24eqeq12d 2625 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
2620, 25rspc2v 3293 . . 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 1254 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818   · cmul 9820  +∞cpnf 9950   ≤ cle 9954  [,)cico 12048  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Ringcrg 18370  AbsValcabv 18639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-abv 18640 This theorem is referenced by:  abv1z  18655  abvneg  18657  abvrec  18659  abvdiv  18660  abvdom  18661  abvres  18662  nmmul  22278  sranlm  22298  abvcxp  25104  qabvexp  25115  ostthlem2  25117  ostth2lem2  25123  ostth3  25127
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