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Theorem isabv 20321
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 20320 . . 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 2820 . 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 6845 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
98eqeq1d 2735 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) = 0 ↔ (πΉβ€˜π‘₯) = 0))
109bibi1d 344 . . . . . 6 (𝑓 = 𝐹 β†’ (((π‘“β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ↔ ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 )))
11 fveq1 6845 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ Β· 𝑦)) = (πΉβ€˜(π‘₯ Β· 𝑦)))
12 fveq1 6845 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
138, 12oveq12d 7379 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) Β· (π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯) Β· (πΉβ€˜π‘¦)))
1411, 13eqeq12d 2749 . . . . . . . 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 6845 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (π‘“β€˜(π‘₯ + 𝑦)) = (πΉβ€˜(π‘₯ + 𝑦)))
168, 12oveq12d 7379 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) + (π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))
1715, 16breq12d 5122 . . . . . . . 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 632 . . . . . . 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 3171 . . . . . 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 632 . . . . 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 3171 . . . 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 3649 . . 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 7394 . . . . 5 (0[,)+∞) ∈ V
242fvexi 6860 . . . . 5 𝐡 ∈ V
2523, 24elmap 8815 . . . 4 (𝐹 ∈ ((0[,)+∞) ↑m 𝐡) ↔ 𝐹:𝐡⟢(0[,)+∞))
2625anbi1i 625 . . 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 275 . 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 287 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   class class class wbr 5109  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑m cmap 8771  0cc0 11059   + caddc 11062   Β· cmul 11064  +∞cpnf 11194   ≀ cle 11198  [,)cico 13275  Basecbs 17091  +gcplusg 17141  .rcmulr 17142  0gc0g 17329  Ringcrg 19972  AbsValcabv 20318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-abv 20319
This theorem is referenced by:  isabvd  20322  abvfge0  20324  abveq0  20328  abvmul  20331  abvtri  20332  abvpropd  20344
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