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Theorem isabv 20278
Description: Elementhood in 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
isabv (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   + (𝑥,𝑦)   · (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isabv
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 abvfval.a . . . 4 𝐴 = (AbsVal‘𝑅)
2 abvfval.b . . . 4 𝐵 = (Base‘𝑅)
3 abvfval.p . . . 4 + = (+g𝑅)
4 abvfval.t . . . 4 · = (.r𝑅)
5 abvfval.z . . . 4 0 = (0g𝑅)
61, 2, 3, 4, 5abvfval 20277 . . 3 (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))})
76eleq2d 2823 . 2 (𝑅 ∈ Ring → (𝐹𝐴𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))}))
8 fveq1 6841 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
98eqeq1d 2738 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0))
109bibi1d 343 . . . . . 6 (𝑓 = 𝐹 → (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹𝑥) = 0 ↔ 𝑥 = 0 )))
11 fveq1 6841 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
12 fveq1 6841 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
138, 12oveq12d 7375 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) · (𝑓𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
1411, 13eqeq12d 2752 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦))))
15 fveq1 6841 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
168, 12oveq12d 7375 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) + (𝑓𝑦)) = ((𝐹𝑥) + (𝐹𝑦)))
1715, 16breq12d 5118 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1814, 17anbi12d 631 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
1918ralbidv 3174 . . . . . 6 (𝑓 = 𝐹 → (∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))) ↔ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
2010, 19anbi12d 631 . . . . 5 (𝑓 = 𝐹 → ((((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2120ralbidv 3174 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦)))) ↔ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2221elrab 3645 . . 3 (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} ↔ (𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
23 ovex 7390 . . . . 5 (0[,)+∞) ∈ V
242fvexi 6856 . . . . 5 𝐵 ∈ V
2523, 24elmap 8809 . . . 4 (𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ↔ 𝐹:𝐵⟶(0[,)+∞))
2625anbi1i 624 . . 3 ((𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))) ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
2722, 26bitri 274 . 2 (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥𝐵 (((𝑓𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) · (𝑓𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓𝑥) + (𝑓𝑦))))} ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥𝐵 (((𝐹𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))))
287, 27bitrdi 286 1 (𝑅 ∈ Ring → (𝐹𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407   class class class wbr 5105  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  0cc0 11051   + caddc 11054   · cmul 11056  +∞cpnf 11186  cle 11190  [,)cico 13266  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  Ringcrg 19964  AbsValcabv 20275
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-abv 20276
This theorem is referenced by:  isabvd  20279  abvfge0  20281  abveq0  20285  abvmul  20288  abvtri  20289  abvpropd  20301
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