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Theorem abveq0 19526
Description: The value of an absolute value is zero iff the argument is zero. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvf.a 𝐴 = (AbsVal‘𝑅)
abvf.b 𝐵 = (Base‘𝑅)
abveq0.z 0 = (0g𝑅)
Assertion
Ref Expression
abveq0 ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))

Proof of Theorem abveq0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . 6 𝐴 = (AbsVal‘𝑅)
21abvrcl 19521 . . . . 5 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . 6 𝐵 = (Base‘𝑅)
4 eqid 2818 . . . . . 6 (+g𝑅) = (+g𝑅)
5 eqid 2818 . . . . . 6 (.r𝑅) = (.r𝑅)
6 abveq0.z . . . . . 6 0 = (0g𝑅)
71, 3, 4, 5, 6isabv 19519 . . . . 5 (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
82, 7syl 17 . . . 4 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 268 . . 3 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
10 simpl 483 . . . 4 ((((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
1110ralimi 3157 . . 3 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
129, 11simpl2im 504 . 2 (𝐹𝐴 → ∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
13 fveqeq2 6672 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) = 0 ↔ (𝐹𝑋) = 0))
14 eqeq1 2822 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
1513, 14bibi12d 347 . . 3 (𝑥 = 𝑋 → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑋) = 0 ↔ 𝑋 = 0 )))
1615rspccva 3619 . 2 ((∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
1712, 16sylan 580 1 ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525   + caddc 10528   · cmul 10530  +∞cpnf 10660  cle 10664  [,)cico 12728  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  0gc0g 16701  Ringcrg 19226  AbsValcabv 19516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-abv 19517
This theorem is referenced by:  abvne0  19527  abv0  19531  abvmet  23112
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