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Theorem abvfval 19027
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 2822 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
54oveq2d 6808 . . . 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 2822 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
98eqeq2d 2780 . . . . . . 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 2822 . . . . . . . . . . 11 (𝑟 = 𝑅 → (.r𝑟) = · )
1413oveqd 6809 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
1514fveq2d 6336 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(.r𝑟)𝑦)) = (𝑓‘(𝑥 · 𝑦)))
1615eqeq1d 2772 . . . . . . . 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 2822 . . . . . . . . . . 11 (𝑟 = 𝑅 → (+g𝑟) = + )
2019oveqd 6809 . . . . . . . . . 10 (𝑟 = 𝑅 → (𝑥(+g𝑟)𝑦) = (𝑥 + 𝑦))
2120fveq2d 6336 . . . . . . . . 9 (𝑟 = 𝑅 → (𝑓‘(𝑥(+g𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
2221breq1d 4794 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑓‘(𝑥(+g𝑟)𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))
2316, 22anbi12d 608 . . . . . . 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 3300 . . . . . 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 608 . . . . 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 3300 . . . 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 3344 . . 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 19026 . . 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 6822 . . . 4 ((0[,)+∞) ↑𝑚 𝐵) ∈ V
3029rabex 4943 . . 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 2816 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 1630  wcel 2144  wral 3060  {crab 3064   class class class wbr 4784  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  0cc0 10137   + caddc 10140   · cmul 10142  +∞cpnf 10272  cle 10276  [,)cico 12381  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  0gc0g 16307  Ringcrg 18754  AbsValcabv 19025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  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 6795  df-abv 19026
This theorem is referenced by:  isabv  19028
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