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Theorem abveq0 18758
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 . . . . . . 7 𝐴 = (AbsVal‘𝑅)
21abvrcl 18753 . . . . . 6 (𝐹𝐴𝑅 ∈ Ring)
3 abvf.b . . . . . . 7 𝐵 = (Base‘𝑅)
4 eqid 2621 . . . . . . 7 (+g𝑅) = (+g𝑅)
5 eqid 2621 . . . . . . 7 (.r𝑅) = (.r𝑅)
6 abveq0.z . . . . . . 7 0 = (0g𝑅)
71, 3, 4, 5, 6isabv 18751 . . . . . 6 (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 . . . . 5 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
98ibi 256 . . . 4 (𝐹𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
109simprd 479 . . 3 (𝐹𝐴 → ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
11 simpl 473 . . . 4 ((((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
1211ralimi 2947 . . 3 (∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))) → ∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
1310, 12syl 17 . 2 (𝐹𝐴 → ∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))
14 fveq2 6153 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1514eqeq1d 2623 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) = 0 ↔ (𝐹𝑋) = 0))
16 eqeq1 2625 . . . 4 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
1715, 16bibi12d 335 . . 3 (𝑥 = 𝑋 → (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑋) = 0 ↔ 𝑋 = 0 )))
1817rspccva 3297 . 2 ((∀𝑥𝐵 ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ 𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
1913, 18sylan 488 1 ((𝐹𝐴𝑋𝐵) → ((𝐹𝑋) = 0 ↔ 𝑋 = 0 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4618  wf 5848  cfv 5852  (class class class)co 6610  0cc0 9888   + caddc 9891   · cmul 9893  +∞cpnf 10023  cle 10027  [,)cico 12127  Basecbs 15792  +gcplusg 15873  .rcmulr 15874  0gc0g 16032  Ringcrg 18479  AbsValcabv 18748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-abv 18749
This theorem is referenced by:  abvne0  18759  abv0  18763  abvmet  22303
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