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Theorem abvmul 19597
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 . . . . . . 7 𝐴 = (AbsVal‘𝑅)
21abvrcl 19589 . . . . . 6 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐵 = (Base‘𝑅)
4 eqid 2801 . . . . . . 7 (+g𝑅) = (+g𝑅)
5 abvmul.t . . . . . . 7 · = (.r𝑅)
6 eqid 2801 . . . . . . 7 (0g𝑅) = (0g𝑅)
71, 3, 4, 5, 6isabv 19587 . . . . . 6 (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 . . . . 5 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 270 . . . 4 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
10 simpl 486 . . . . . . 7 (((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1110ralimi 3131 . . . . . 6 (∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))) → ∀𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1211adantl 485 . . . . 5 ((((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1312ralimi 3131 . . . 4 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
149, 13simpl2im 507 . . 3 (𝐹𝐴 → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
15 fvoveq1 7162 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑋 · 𝑦)))
16 fveq2 6649 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1716oveq1d 7154 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑦)))
1815, 17eqeq12d 2817 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦))))
19 oveq2 7147 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
2019fveq2d 6653 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 · 𝑦)) = (𝐹‘(𝑋 · 𝑌)))
21 fveq2 6649 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
2221oveq2d 7155 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) · (𝐹𝑦)) = ((𝐹𝑋) · (𝐹𝑌)))
2320, 22eqeq12d 2817 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 · 𝑦)) = ((𝐹𝑋) · (𝐹𝑦)) ↔ (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
2418, 23rspc2v 3584 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
2514, 24syl5com 31 . 2 (𝐹𝐴 → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌))))
26253impib 1113 1 ((𝐹𝐴𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹𝑋) · (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109   class class class wbr 5033  wf 6324  cfv 6328  (class class class)co 7139  0cc0 10530   + caddc 10533   · cmul 10535  +∞cpnf 10665  cle 10669  [,)cico 12732  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  0gc0g 16709  Ringcrg 19294  AbsValcabv 19584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-abv 19585
This theorem is referenced by:  abv1z  19600  abvneg  19602  abvrec  19604  abvdiv  19605  abvdom  19606  abvres  19607  nmmul  23274  sranlm  23294  abvcxp  26203  qabvexp  26214  ostthlem2  26216  ostth2lem2  26222  ostth3  26226
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