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Theorem abveq0 20433
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 = (AbsValβ€˜π‘…)
abvf.b 𝐡 = (Baseβ€˜π‘…)
abveq0.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
abveq0 ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((πΉβ€˜π‘‹) = 0 ↔ 𝑋 = 0 ))

Proof of Theorem abveq0
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvf.a . . . . . 6 𝐴 = (AbsValβ€˜π‘…)
21abvrcl 20428 . . . . 5 (𝐹 ∈ 𝐴 β†’ 𝑅 ∈ Ring)
3 abvf.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
4 eqid 2732 . . . . . 6 (+gβ€˜π‘…) = (+gβ€˜π‘…)
5 eqid 2732 . . . . . 6 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6 abveq0.z . . . . . 6 0 = (0gβ€˜π‘…)
71, 3, 4, 5, 6isabv 20426 . . . . 5 (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
82, 7syl 17 . . . 4 (𝐹 ∈ 𝐴 β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
98ibi 266 . . 3 (𝐹 ∈ 𝐴 β†’ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
10 simpl 483 . . . 4 ((((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
1110ralimi 3083 . . 3 (βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯(.rβ€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯(+gβ€˜π‘…)𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))) β†’ βˆ€π‘₯ ∈ 𝐡 ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
129, 11simpl2im 504 . 2 (𝐹 ∈ 𝐴 β†’ βˆ€π‘₯ ∈ 𝐡 ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ))
13 fveqeq2 6900 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) = 0 ↔ (πΉβ€˜π‘‹) = 0))
14 eqeq1 2736 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ = 0 ↔ 𝑋 = 0 ))
1513, 14bibi12d 345 . . 3 (π‘₯ = 𝑋 β†’ (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((πΉβ€˜π‘‹) = 0 ↔ 𝑋 = 0 )))
1615rspccva 3611 . 2 ((βˆ€π‘₯ ∈ 𝐡 ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ 𝑋 ∈ 𝐡) β†’ ((πΉβ€˜π‘‹) = 0 ↔ 𝑋 = 0 ))
1712, 16sylan 580 1 ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) β†’ ((πΉβ€˜π‘‹) = 0 ↔ 𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  0cc0 11109   + caddc 11112   Β· cmul 11114  +∞cpnf 11244   ≀ cle 11248  [,)cico 13325  Basecbs 17143  +gcplusg 17196  .rcmulr 17197  0gc0g 17384  Ringcrg 20055  AbsValcabv 20423
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-abv 20424
This theorem is referenced by:  abvne0  20434  abv0  20438  abvmet  24083
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