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Theorem abvfval 20569
Description: Value of the set of absolute values. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypotheses
Ref Expression
abvfval.a 𝐴 = (AbsValβ€˜π‘…)
abvfval.b 𝐡 = (Baseβ€˜π‘…)
abvfval.p + = (+gβ€˜π‘…)
abvfval.t Β· = (.rβ€˜π‘…)
abvfval.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
abvfval (𝑅 ∈ Ring β†’ 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
Distinct variable groups:   π‘₯,𝑓,𝑦,𝐡   + ,𝑓   𝑅,𝑓,π‘₯,𝑦   Β· ,𝑓   0 ,𝑓
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑓)   + (π‘₯,𝑦)   Β· (π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem abvfval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 abvfval.a . 2 𝐴 = (AbsValβ€˜π‘…)
2 fveq2 6891 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
3 abvfval.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
42, 3eqtr4di 2790 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
54oveq2d 7427 . . . 4 (π‘Ÿ = 𝑅 β†’ ((0[,)+∞) ↑m (Baseβ€˜π‘Ÿ)) = ((0[,)+∞) ↑m 𝐡))
6 fveq2 6891 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
7 abvfval.z . . . . . . . . 9 0 = (0gβ€˜π‘…)
86, 7eqtr4di 2790 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = 0 )
98eqeq2d 2743 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (π‘₯ = (0gβ€˜π‘Ÿ) ↔ π‘₯ = 0 ))
109bibi2d 342 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ↔ ((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 )))
11 fveq2 6891 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
12 abvfval.t . . . . . . . . . . 11 Β· = (.rβ€˜π‘…)
1311, 12eqtr4di 2790 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
1413oveqd 7428 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘₯(.rβ€˜π‘Ÿ)𝑦) = (π‘₯ Β· 𝑦))
1514fveqeq2d 6899 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ↔ (π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦))))
16 fveq2 6891 . . . . . . . . . . . 12 (π‘Ÿ = 𝑅 β†’ (+gβ€˜π‘Ÿ) = (+gβ€˜π‘…))
17 abvfval.p . . . . . . . . . . . 12 + = (+gβ€˜π‘…)
1816, 17eqtr4di 2790 . . . . . . . . . . 11 (π‘Ÿ = 𝑅 β†’ (+gβ€˜π‘Ÿ) = + )
1918oveqd 7428 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (π‘₯(+gβ€˜π‘Ÿ)𝑦) = (π‘₯ + 𝑦))
2019fveq2d 6895 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) = (π‘“β€˜(π‘₯ + 𝑦)))
2120breq1d 5158 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ ((π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)) ↔ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))
2215, 21anbi12d 631 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))) ↔ ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)))))
234, 22raleqbidv 3342 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))) ↔ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)))))
2410, 23anbi12d 631 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)))) ↔ (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))))
254, 24raleqbidv 3342 . . . 4 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)(((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)))) ↔ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))))
265, 25rabeqbidv 3449 . . 3 (π‘Ÿ = 𝑅 β†’ {𝑓 ∈ ((0[,)+∞) ↑m (Baseβ€˜π‘Ÿ)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)(((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))} = {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
27 df-abv 20568 . . 3 AbsVal = (π‘Ÿ ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Baseβ€˜π‘Ÿ)) ∣ βˆ€π‘₯ ∈ (Baseβ€˜π‘Ÿ)(((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = (0gβ€˜π‘Ÿ)) ∧ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘“β€˜(π‘₯(.rβ€˜π‘Ÿ)𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯(+gβ€˜π‘Ÿ)𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
28 ovex 7444 . . . 4 ((0[,)+∞) ↑m 𝐡) ∈ V
2928rabex 5332 . . 3 {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))} ∈ V
3026, 27, 29fvmpt 6998 . 2 (𝑅 ∈ Ring β†’ (AbsValβ€˜π‘…) = {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
311, 30eqtrid 2784 1 (𝑅 ∈ Ring β†’ 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐡) ∣ βˆ€π‘₯ ∈ 𝐡 (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411   ↑m cmap 8822  0cc0 11112   + caddc 11115   Β· cmul 11117  +∞cpnf 11249   ≀ cle 11253  [,)cico 13330  Basecbs 17148  +gcplusg 17201  .rcmulr 17202  0gc0g 17389  Ringcrg 20127  AbsValcabv 20567
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-abv 20568
This theorem is referenced by:  isabv  20570
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