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Theorem abveq0 20426
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 20421 . . . . 5 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . 6 𝐵 = (Base‘𝑅)
4 eqid 2732 . . . . . 6 (+g𝑅) = (+g𝑅)
5 eqid 2732 . . . . . 6 (.r𝑅) = (.r𝑅)
6 abveq0.z . . . . . 6 0 = (0g𝑅)
71, 3, 4, 5, 6isabv 20419 . . . . 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 266 . . 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 3083 . . 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 6897 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) = 0 ↔ (𝐹𝑋) = 0))
14 eqeq1 2736 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
1513, 14bibi12d 345 . . 3 (𝑥 = 𝑋 → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑋) = 0 ↔ 𝑋 = 0 )))
1615rspccva 3611 . 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 205  wa 396   = wceq 1541  wcel 2106  wral 3061   class class class wbr 5147  wf 6536  cfv 6540  (class class class)co 7405  0cc0 11106   + caddc 11109   · cmul 11111  +∞cpnf 11241  cle 11245  [,)cico 13322  Basecbs 17140  +gcplusg 17193  .rcmulr 17194  0gc0g 17381  Ringcrg 20049  AbsValcabv 20416
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-abv 20417
This theorem is referenced by:  abvne0  20427  abv0  20431  abvmet  24075
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