Theorem abvmul 19599

Description: An absolute value distributes under multiplication. (Contributed by Mario Carneiro, 8-Sep-2014.)

Hypotheses

Ref	Expression
abvf.a	⊢ 𝐴 = (AbsVal‘𝑅)
abvf.b	⊢ 𝐵 = (Base‘𝑅)
abvmul.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
abvmul	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌)))

Proof of Theorem abvmul

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abvf.a	. . . . . . 7 ⊢ 𝐴 = (AbsVal‘𝑅)
2	1	abvrcl 19591	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)
3		abvf.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2821	. . . . . . 7 ⊢ (+g‘𝑅) = (+g‘𝑅)
5		abvmul.t	. . . . . . 7 ⊢ · = (.r‘𝑅)
6		eqid 2821	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	1, 3, 4, 5, 6	isabv 19589	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
8	2, 7	syl 17	. . . . 5 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
9	8	ibi 269	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
10		simpl 485	. . . . . . 7 ⊢ (((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
11	10	ralimi 3160	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
12	11	adantl 484	. . . . 5 ⊢ ((((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))) → ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
13	12	ralimi 3160	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
14	9, 13	simpl2im 506	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
15		fvoveq1 7178	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑋 · 𝑦)))
16		fveq2 6669	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
17	16	oveq1d 7170	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑋) · (𝐹‘𝑦)))
18	15, 17	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 · 𝑦)) = ((𝐹‘𝑋) · (𝐹‘𝑦))))
19		oveq2 7163	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
20	19	fveq2d 6673	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 · 𝑦)) = (𝐹‘(𝑋 · 𝑌)))
21		fveq2 6669	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
22	21	oveq2d 7171	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) · (𝐹‘𝑦)) = ((𝐹‘𝑋) · (𝐹‘𝑌)))
23	20, 22	eqeq12d 2837	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 · 𝑦)) = ((𝐹‘𝑋) · (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌))))
24	18, 23	rspc2v 3632	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌))))
25	14, 24	syl5com 31	. 2 ⊢ (𝐹 ∈ 𝐴 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌))))
26	25	3impib 1112	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 · 𝑌)) = ((𝐹‘𝑋) · (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138   class class class wbr 5065  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  0cc0 10536   + caddc 10539   · cmul 10541  +∞cpnf 10671   ≤ cle 10675  [,)cico 12739  Basecbs 16482  +_gcplusg 16564  ._rcmulr 16565  0_gc0g 16712  Ringcrg 19296  AbsValcabv 19586

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8407  df-abv 19587

This theorem is referenced by:  abv1z  19602  abvneg  19604  abvrec  19606  abvdiv  19607  abvdom  19608  abvres  19609  nmmul  23272  sranlm  23292  abvcxp  26190  qabvexp  26201  ostthlem2  26203  ostth2lem2  26209  ostth3  26213