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Theorem abveq0 19716
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 19711 . . . . 5 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . 6 𝐵 = (Base‘𝑅)
4 eqid 2738 . . . . . 6 (+g𝑅) = (+g𝑅)
5 eqid 2738 . . . . . 6 (.r𝑅) = (.r𝑅)
6 abveq0.z . . . . . 6 0 = (0g𝑅)
71, 3, 4, 5, 6isabv 19709 . . . . 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 270 . . 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 486 . . . 4 ((((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
1110ralimi 3075 . . 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 507 . 2 (𝐹𝐴 → ∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
13 fveqeq2 6683 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) = 0 ↔ (𝐹𝑋) = 0))
14 eqeq1 2742 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
1513, 14bibi12d 349 . . 3 (𝑥 = 𝑋 → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑋) = 0 ↔ 𝑋 = 0 )))
1615rspccva 3525 . 2 ((∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
1712, 16sylan 583 1 ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3053   class class class wbr 5030  wf 6335  cfv 6339  (class class class)co 7170  0cc0 10615   + caddc 10618   · cmul 10620  +∞cpnf 10750  cle 10754  [,)cico 12823  Basecbs 16586  +gcplusg 16668  .rcmulr 16669  0gc0g 16816  Ringcrg 19416  AbsValcabv 19706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-abv 19707
This theorem is referenced by:  abvne0  19717  abv0  19721  abvmet  23328
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