abvfval - Metamath Proof Explorer

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Theorem abvfval 20426

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 = (0_g‘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 6892	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		abvfval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5	4	oveq2d 7425	. . . 4 ⊢ (𝑟 = 𝑅 → ((0[,)+∞) ↑_m (Base‘𝑟)) = ((0[,)+∞) ↑_m 𝐵))
6		fveq2 6892	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0_g‘𝑟) = (0_g‘𝑅))
7		abvfval.z	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑅)
8	6, 7	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0_g‘𝑟) = 0 )
9	8	eqeq2d 2744	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑥 = (0_g‘𝑟) ↔ 𝑥 = 0 ))
10	9	bibi2d 343	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑓‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑟)) ↔ ((𝑓‘𝑥) = 0 ↔ 𝑥 = 0 )))
11		fveq2 6892	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (._r‘𝑟) = (._r‘𝑅))
12		abvfval.t	. . . . . . . . . . 11 ⊢ · = (._r‘𝑅)
13	11, 12	eqtr4di 2791	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (._r‘𝑟) = · )
14	13	oveqd 7426	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑥(._r‘𝑟)𝑦) = (𝑥 · 𝑦))
15	14	fveqeq2d 6900	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((𝑓‘(𝑥(._r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ↔ (𝑓‘(𝑥 · 𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦))))
16		fveq2 6892	. . . . . . . . . . . 12 ⊢ (𝑟 = 𝑅 → (+_g‘𝑟) = (+_g‘𝑅))
17		abvfval.p	. . . . . . . . . . . 12 ⊢ + = (+_g‘𝑅)
18	16, 17	eqtr4di 2791	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑅 → (+_g‘𝑟) = + )
19	18	oveqd 7426	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (𝑥(+_g‘𝑟)𝑦) = (𝑥 + 𝑦))
20	19	fveq2d 6896	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝑓‘(𝑥(+_g‘𝑟)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
21	20	breq1d 5159	. . . . . . . 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 3343	. . . . . 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 = (0_g‘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 3343	. . . 4 ⊢ (𝑟 = 𝑅 → (∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0_g‘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 3450	. . 3 ⊢ (𝑟 = 𝑅 → {𝑓 ∈ ((0[,)+∞) ↑_m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0_g‘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 20425	. . 3 ⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑_m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0_g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(._r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+_g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))})
28		ovex 7442	. . . 4 ⊢ ((0[,)+∞) ↑_m 𝐵) ∈ V
29	28	rabex 5333	. . 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 6999	. 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 2785	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))))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 0cc0 11110 + caddc 11113 · cmul 11115 +∞cpnf 11245 ≤ cle 11249 [,)cico 13326 Basecbs 17144 +_gcplusg 17197 ._rcmulr 17198 0_gc0g 17385 Ringcrg 20056 AbsValcabv 20424

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 5300 ax-nul 5307 ax-pr 5428

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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-abv 20425

This theorem is referenced by: isabv 20427

W3C validator