Theorem abvfval 20828

Description: Value of 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
abvfval	⊢ (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))))})

Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   + ,𝑓   𝑅,𝑓,𝑥,𝑦   · ,𝑓   0 ,𝑓

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑓)   + (𝑥,𝑦)   · (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem abvfval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abvfval.a	. 2 ⊢ 𝐴 = (AbsVal‘𝑅)
2		fveq2 6907	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		abvfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2793	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	oveq2d 7447	. . . 4 ⊢ (𝑟 = 𝑅 → ((0[,)+∞) ↑m (Base‘𝑟)) = ((0[,)+∞) ↑m 𝐵))
6		fveq2 6907	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
7		abvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
8	6, 7	eqtr4di 2793	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
9	8	eqeq2d 2746	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 = (0g‘𝑟) ↔ 𝑥 = 0 ))
10	9	bibi2d 342	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ↔ ((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 )))
11		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
12		abvfval.t	. . . . . . . . . . 11 ⊢ · = (.r‘𝑅)
13	11, 12	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
14	13	oveqd 7448	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
15	14	fveqeq2d 6915	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦))))
16		fveq2 6907	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = (+g‘𝑅))
17		abvfval.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑅)
18	16, 17	eqtr4di 2793	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (+g‘𝑟) = + )
19	18	oveqd 7448	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(+g‘𝑟)𝑦) = (𝑥 + 𝑦))
20	19	fveq2d 6911	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓‘(𝑥(+g‘𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
21	20	breq1d 5158	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))
22	15, 21	anbi12d 632	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))
23	4, 22	raleqbidv 3344	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥 + 𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦)))))
24	10, 23	anbi12d 632	. . . . 5 ⊢ (𝑟 = 𝑅 → ((((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)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))))))
25	4, 24	raleqbidv 3344	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)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))))))
26	5, 25	rabeqbidv 3452	. . 3 ⊢ (𝑟 = 𝑅 → {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)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))))})
27		df-abv 20827	. . 3 ⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
28		ovex 7464	. . . 4 ⊢ ((0[,)+∞) ↑m 𝐵) ∈ V
29	28	rabex 5345	. . 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))))} ∈ V
30	26, 27, 29	fvmpt 7016	. 2 ⊢ (𝑅 ∈ Ring → (AbsVal‘𝑅) = {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))))})
31	1, 30	eqtrid 2787	1 ⊢ (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))))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  0cc0 11153   + caddc 11156   · cmul 11158  +∞cpnf 11290   ≤ cle 11294  [,)cico 13386  Basecbs 17245  +_gcplusg 17298  ._rcmulr 17299  0_gc0g 17486  Ringcrg 20251  AbsValcabv 20826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-abv 20827

This theorem is referenced by:  isabv  20829