Theorem isabv 21080

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	⊢ + = (+g‘𝑅)
abvfval.t	⊢ · = (.r‘𝑅)
abvfval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
isabv	⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))

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.

Step	Hyp	Ref	Expression
1		abvfval.a	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
2		abvfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
3		abvfval.p	. . . 4 ⊢ + = (+g‘𝑅)
4		abvfval.t	. . . 4 ⊢ · = (.r‘𝑅)
5		abvfval.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6	1, 2, 3, 4, 5	abvfval 21079	. . 3 ⊢ (𝑅 ∈ Ring → 𝐴 = {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
7	6	eleq2d 2849	. 2 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ 𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}))
8		fveq1 6892	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
9	8	eqeq1d 2765	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) = 0 ↔ (𝐹‘𝑥) = 0))
10	9	bibi1d 346	. . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ↔ ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 )))
11		fveq1 6892	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 · 𝑦)) = (𝐹‘(𝑥 · 𝑦)))
12		fveq1 6892	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	8, 12	oveq12d 7445	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) · (𝑓‘𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
14	11, 13	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦))))
15		fveq1 6892	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
16	8, 12	oveq12d 7445	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) + (𝑓‘𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
17	15, 16	breq12d 5118	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
18	14, 17	anbi12d 644	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))
19	18	ralbidv 3188	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))
20	10, 19	anbi12d 644	. . . . 5 ⊢ (𝑓 = 𝐹 → ((((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
21	20	ralbidv 3188	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))) ↔ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
22	21	elrab 3647	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))} ↔ (𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
23		ovex 7460	. . . . 5 ⊢ (0[,)+∞) ∈ V
24	2	fvexi 6907	. . . . 5 ⊢ 𝐵 ∈ V
25	23, 24	elmap 8910	. . . 4 ⊢ (𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ↔ 𝐹:𝐵⟶(0[,)+∞))
26	25	anbi1i 636	. . 3 ⊢ ((𝐹 ∈ ((0[,)+∞) ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))) ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
27	22, 26	bitri 278	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ ((0[,)+∞) ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 (((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))} ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
28	7, 27	bitrdi 290	1 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079  {crab 3415   class class class wbr 5105  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427   ↑_m cmap 8858  0cc0 11210   + caddc 11213   · cmul 11215  +∞cpnf 11350   ≤ cle 11354  [,)cico 13488  Basecbs 17399  +_gcplusg 17440  ._rcmulr 17441  0_gc0g 17622  Ringcrg 20471  AbsValcabv 21077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-abv 21078

This theorem is used by:  isabvd  21081  abvfge0  21083  abveq0  21087  abvmul  21090  abvtri  21091  abvpropd  21104