Theorem abvtri 19594

Description: An absolute value satisfies the triangle inequality. (Contributed by Mario Carneiro, 8-Sep-2014.)

Hypotheses

Ref	Expression
abvf.a	⊢ 𝐴 = (AbsVal‘𝑅)
abvf.b	⊢ 𝐵 = (Base‘𝑅)
abvtri.p	⊢ + = (+g‘𝑅)

Assertion

Ref	Expression
abvtri	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))

Proof of Theorem abvtri

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 19585	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)
3		abvf.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4		abvtri.p	. . . . . . 7 ⊢ + = (+g‘𝑅)
5		eqid 2798	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2798	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	1, 3, 4, 5, 6	isabv 19583	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 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(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
9	8	ibi 270	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))
10		simpr 488	. . . . . . 7 ⊢ (((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
11	10	ralimi 3128	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) → ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
12	11	adantl 485	. . . . 5 ⊢ ((((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))) → ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
13	12	ralimi 3128	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
14	9, 13	simpl2im 507	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
15		fvoveq1 7158	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
16		fveq2 6645	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
17	16	oveq1d 7150	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘𝑋) + (𝐹‘𝑦)))
18	15, 17	breq12d 5043	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑦))))
19		oveq2 7143	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
20	19	fveq2d 6649	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
21		fveq2 6645	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
22	21	oveq2d 7151	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) + (𝐹‘𝑦)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))
23	20, 22	breq12d 5043	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌))))
24	18, 23	rspc2v 3581	. . 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 1113	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  0cc0 10526   + caddc 10529   · cmul 10531  +∞cpnf 10661   ≤ cle 10665  [,)cico 12728  Basecbs 16475  +_gcplusg 16557  ._rcmulr 16558  0_gc0g 16705  Ringcrg 19290  AbsValcabv 19580

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-abv 19581

This theorem is referenced by:  abvsubtri  19599  abvres  19603  abvcxp  26199  qabvle  26209  ostth2lem2  26218  ostth3  26222