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Theorem abvfval 18641
 Description: Value of 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
abvfval (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   + ,𝑓   𝑅,𝑓,𝑥,𝑦   · ,𝑓   0 ,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)   + (𝑥,𝑦)   · (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem abvfval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 abvfval.a . 2 𝐴 = (AbsVal‘𝑅)
2 fveq2 6103 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 abvfval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2662 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54oveq2d 6565 . . . 4 (𝑟 = 𝑅 → ((0[,)+∞) ↑𝑚 (Base‘𝑟)) = ((0[,)+∞) ↑𝑚 𝐵))
6 fveq2 6103 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 abvfval.z . . . . . . . . 9 0 = (0g𝑅)
86, 7syl6eqr 2662 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98eqeq2d 2620 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
109bibi2d 331 . . . . . 6 (𝑟 = 𝑅 → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = 0 )))
11 fveq2 6103 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
12 abvfval.t . . . . . . . . . . . 12 · = (.r𝑅)
1311, 12syl6eqr 2662 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqd 6566 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
1514fveq2d 6107 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(.r𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
1615eqeq1d 2612 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
17 fveq2 6103 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (+g𝑟) = (+g𝑅))
18 abvfval.p . . . . . . . . . . . 12 + = (+g𝑅)
1917, 18syl6eqr 2662 . . . . . . . . . . 11 (𝑟 = 𝑅 → (+g𝑟) = + )
2019oveqd 6566 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(+g𝑟)𝑦) = (𝑥 + 𝑦))
2120fveq2d 6107 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(+g𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
2221breq1d 4593 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
2316, 22anbi12d 743 . . . . . . 7 (𝑟 = 𝑅 → (((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
244, 23raleqbidv 3129 . . . . . 6 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))))
2510, 24anbi12d 743 . . . . 5 (𝑟 = 𝑅 → ((((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
264, 25raleqbidv 3129 . . . 4 (𝑟 = 𝑅 → (∀𝑥 ∈ (Base‘𝑟)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))))
275, 26rabeqbidv 3168 . . 3 (𝑟 = 𝑅 → {𝑓 ∈ ((0[,)+∞) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} = {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
28 df-abv 18640 . . 3 AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑𝑚 (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
29 ovex 6577 . . . 4 ((0[,)+∞) ↑𝑚 𝐵) ∈ V
3029rabex 4740 . . 3 {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} ∈ V
3127, 28, 30fvmpt 6191 . 2 (𝑅 ∈ Ring → (AbsVal‘𝑅) = {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
321, 31syl5eq 2656 1 (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑𝑚 𝐵) ∣ ∀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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  0cc0 9815   + caddc 9818   · cmul 9820  +∞cpnf 9950   ≤ cle 9954  [,)cico 12048  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Ringcrg 18370  AbsValcabv 18639 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-abv 18640 This theorem is referenced by:  isabv  18642
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