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Theorem isabv 19585
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 19584 . . 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 2901 . 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 6658 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2826 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0))
109bibi1d 347 . . . . . 6 (𝑓 = 𝐹 → (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑥) = 0 ↔ 𝑥 = 0 )))
11 fveq1 6658 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
12 fveq1 6658 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
138, 12oveq12d 7164 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) · (𝑓𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1411, 13eqeq12d 2840 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦))))
15 fveq1 6658 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
168, 12oveq12d 7164 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
1715, 16breq12d 5066 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1814, 17anbi12d 633 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
1918ralbidv 3192 . . . . . 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 633 . . . . 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 3192 . . . 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 3666 . . 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 7179 . . . . 5 (0[,)+∞) ∈ V
242fvexi 6673 . . . . 5 𝐵 ∈ V
2523, 24elmap 8427 . . . 4 (𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ↔ 𝐹:𝐵⟶(0[,)+∞))
2625anbi1i 626 . . 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))))))
2722, 26bitri 278 . 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))))))
287, 27syl6bb 290 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 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137   class class class wbr 5053  wf 6340  cfv 6344  (class class class)co 7146  m cmap 8398  0cc0 10531   + caddc 10534   · cmul 10536  +∞cpnf 10666  cle 10670  [,)cico 12735  Basecbs 16481  +gcplusg 16563  .rcmulr 16564  0gc0g 16711  Ringcrg 19295  AbsValcabv 19582
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400  df-abv 19583
This theorem is referenced by:  isabvd  19586  abvfge0  19588  abveq0  19592  abvmul  19595  abvtri  19596  abvpropd  19608
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