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Theorem isabv 20570
Description: Elementhood in 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
isabv (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐹,𝑦   π‘₯,𝑅,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   + (π‘₯,𝑦)   Β· (π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem isabv
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 abvfval.a . . . 4 𝐴 = (AbsValβ€˜π‘…)
2 abvfval.b . . . 4 𝐡 = (Baseβ€˜π‘…)
3 abvfval.p . . . 4 + = (+gβ€˜π‘…)
4 abvfval.t . . . 4 Β· = (.rβ€˜π‘…)
5 abvfval.z . . . 4 0 = (0gβ€˜π‘…)
61, 2, 3, 4, 5abvfval 20569 . . 3 (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‘¦))))})
76eleq2d 2819 . 2 (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ {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‘¦))))}))
8 fveq1 6890 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
98eqeq1d 2734 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) = 0 ↔ (πΉβ€˜π‘₯) = 0))
109bibi1d 343 . . . . . 6 (𝑓 = 𝐹 β†’ (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 )))
11 fveq1 6890 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ Β· 𝑦)) = (πΉβ€˜(π‘₯ Β· 𝑦)))
12 fveq1 6890 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
138, 12oveq12d 7429 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
1411, 13eqeq12d 2748 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ↔ (πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦))))
15 fveq1 6890 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ + 𝑦)) = (πΉβ€˜(π‘₯ + 𝑦)))
168, 12oveq12d 7429 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
1715, 16breq12d 5161 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)) ↔ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))
1814, 17anbi12d 631 . . . . . . 7 (𝑓 = 𝐹 β†’ (((π‘“β€˜(π‘₯ Β· 𝑦)) = ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) ∧ (π‘“β€˜(π‘₯ + 𝑦)) ≀ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦))) ↔ ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))
1918ralbidv 3177 . . . . . 6 (𝑓 = 𝐹 β†’ (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‘¦))) ↔ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))))
2010, 19anbi12d 631 . . . . 5 (𝑓 = 𝐹 β†’ ((((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‘¦)))) ↔ (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
2120ralbidv 3177 . . . 4 (𝑓 = 𝐹 β†’ (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‘¦)))) ↔ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
2221elrab 3683 . . 3 (𝐹 ∈ {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‘¦))))} ↔ (𝐹 ∈ ((0[,)+∞) ↑m 𝐡) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
23 ovex 7444 . . . . 5 (0[,)+∞) ∈ V
242fvexi 6905 . . . . 5 𝐡 ∈ V
2523, 24elmap 8867 . . . 4 (𝐹 ∈ ((0[,)+∞) ↑m 𝐡) ↔ 𝐹:𝐡⟢(0[,)+∞))
2625anbi1i 624 . . 3 ((𝐹 ∈ ((0[,)+∞) ↑m 𝐡) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))) ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
2722, 26bitri 274 . 2 (𝐹 ∈ {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‘¦))))} ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦))))))
287, 27bitrdi 286 1 (𝑅 ∈ Ring β†’ (𝐹 ∈ 𝐴 ↔ (𝐹:𝐡⟢(0[,)+∞) ∧ βˆ€π‘₯ ∈ 𝐡 (((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝐡 ((πΉβ€˜(π‘₯ Β· 𝑦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)) ∧ (πΉβ€˜(π‘₯ + 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜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  βŸΆwf 6539  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-pow 5363  ax-pr 5427  ax-un 7727
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-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  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-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-abv 20568
This theorem is referenced by:  isabvd  20571  abvfge0  20573  abveq0  20577  abvmul  20580  abvtri  20581  abvpropd  20593
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