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Theorem abvfval 19028
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 6332 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 abvfval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2823 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54oveq2d 6809 . . . 4 (𝑟 = 𝑅 → ((0[,)+∞) ↑𝑚 (Base‘𝑟)) = ((0[,)+∞) ↑𝑚 𝐵))
6 fveq2 6332 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
7 abvfval.z . . . . . . . . 9 0 = (0g𝑅)
86, 7syl6eqr 2823 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98eqeq2d 2781 . . . . . . 7 (𝑟 = 𝑅 → (𝑥 = (0g𝑟) ↔ 𝑥 = 0 ))
109bibi2d 331 . . . . . 6 (𝑟 = 𝑅 → (((𝑓𝑥) = 0 ↔ 𝑥 = (0g𝑟)) ↔ ((𝑓𝑥) = 0 ↔ 𝑥 = 0 )))
11 fveq2 6332 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
12 abvfval.t . . . . . . . . . . . 12 · = (.r𝑅)
1311, 12syl6eqr 2823 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqd 6810 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
1514fveq2d 6336 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(.r𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
1615eqeq1d 2773 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦))))
17 fveq2 6332 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (+g𝑟) = (+g𝑅))
18 abvfval.p . . . . . . . . . . . 12 + = (+g𝑅)
1917, 18syl6eqr 2823 . . . . . . . . . . 11 (𝑟 = 𝑅 → (+g𝑟) = + )
2019oveqd 6810 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(+g𝑟)𝑦) = (𝑥 + 𝑦))
2120fveq2d 6336 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(+g𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
2221breq1d 4796 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
2316, 22anbi12d 616 . . . . . . 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 3301 . . . . . 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 616 . . . . 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 3301 . . . 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 3345 . . 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 19027 . . 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 6823 . . . 4 ((0[,)+∞) ↑𝑚 𝐵) ∈ V
3029rabex 4946 . . 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 6424 . 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 2817 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061  {crab 3065   class class class wbr 4786  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  0cc0 10138   + caddc 10141   · cmul 10143  +∞cpnf 10273  cle 10277  [,)cico 12382  Basecbs 16064  +gcplusg 16149  .rcmulr 16150  0gc0g 16308  Ringcrg 18755  AbsValcabv 19026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-abv 19027
This theorem is referenced by:  isabv  19029
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