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Theorem isabv 18867
Description: Elementhood in the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvfval.a 𝐴 = (AbsVal‘𝑅)
abvfval.b 𝐵 = (Base‘𝑅)
abvfval.p + = (+g𝑅)
abvfval.t · = (.r𝑅)
abvfval.z 0 = (0g𝑅)
Assertion
Ref Expression
isabv (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   + (𝑥,𝑦)   · (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isabv
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 abvfval.a . . . 4 𝐴 = (AbsVal‘𝑅)
2 abvfval.b . . . 4 𝐵 = (Base‘𝑅)
3 abvfval.p . . . 4 + = (+g𝑅)
4 abvfval.t . . . 4 · = (.r𝑅)
5 abvfval.z . . . 4 0 = (0g𝑅)
61, 2, 3, 4, 5abvfval 18866 . . 3 (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
76eleq2d 2716 . 2 (𝑅 ∈ Ring → (𝐹𝐴𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))}))
8 fveq1 6228 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2653 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0))
109bibi1d 332 . . . . . 6 (𝑓 = 𝐹 → (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑥) = 0 ↔ 𝑥 = 0 )))
11 fveq1 6228 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
12 fveq1 6228 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
138, 12oveq12d 6708 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) · (𝑓𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1411, 13eqeq12d 2666 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦))))
15 fveq1 6228 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
168, 12oveq12d 6708 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
1715, 16breq12d 4698 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1814, 17anbi12d 747 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
1918ralbidv 3015 . . . . . 6 (𝑓 = 𝐹 → (∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
2010, 19anbi12d 747 . . . . 5 (𝑓 = 𝐹 → ((((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2120ralbidv 3015 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2221elrab 3396 . . 3 (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} ↔ (𝐹 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
23 ovex 6718 . . . . 5 (0[,)+∞) ∈ V
24 fvex 6239 . . . . . 6 (Base‘𝑅) ∈ V
252, 24eqeltri 2726 . . . . 5 𝐵 ∈ V
2623, 25elmap 7928 . . . 4 (𝐹 ∈ ((0[,)+∞) ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶(0[,)+∞))
2726anbi1i 731 . . 3 ((𝐹 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))) ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2822, 27bitri 264 . 2 (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
297, 28syl6bb 276 1 (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974   + caddc 9977   · cmul 9979  +∞cpnf 10109  cle 10113  [,)cico 12215  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  0gc0g 16147  Ringcrg 18593  AbsValcabv 18864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-abv 18865
This theorem is referenced by:  isabvd  18868  abvfge0  18870  abveq0  18874  abvmul  18877  abvtri  18878  abvpropd  18890
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