Theorem abvtri 19603

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 19594	. . . . . 6 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring)
3		abvf.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
4		abvtri.p	. . . . . . 7 ⊢ + = (+g‘𝑅)
5		eqid 2823	. . . . . . 7 ⊢ (.r‘𝑅) = (.r‘𝑅)
6		eqid 2823	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
7	1, 3, 4, 5, 6	isabv 19592	. . . . . 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 269	. . . 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 487	. . . . . . 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 3162	. . . . . 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 484	. . . . 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 3162	. . . 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 506	. . 3 ⊢ (𝐹 ∈ 𝐴 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
15		fvoveq1 7181	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
16		fveq2 6672	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
17	16	oveq1d 7173	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) + (𝐹‘𝑦)) = ((𝐹‘𝑋) + (𝐹‘𝑦)))
18	15, 17	breq12d 5081	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑦))))
19		oveq2 7166	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
20	19	fveq2d 6676	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
21		fveq2 6672	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
22	21	oveq2d 7174	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) + (𝐹‘𝑦)) = ((𝐹‘𝑋) + (𝐹‘𝑌)))
23	20, 22	breq12d 5081	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) ≤ ((𝐹‘𝑋) + (𝐹‘𝑌))))
24	18, 23	rspc2v 3635	. . 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 1537   ∈ wcel 2114  ∀wral 3140   class class class wbr 5068  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  0cc0 10539   + caddc 10542   · cmul 10544  +∞cpnf 10674   ≤ cle 10678  [,)cico 12743  Basecbs 16485  +_gcplusg 16567  ._rcmulr 16568  0_gc0g 16715  Ringcrg 19299  AbsValcabv 19589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-abv 19590

This theorem is referenced by:  abvsubtri  19608  abvres  19612  abvcxp  26193  qabvle  26203  ostth2lem2  26212  ostth3  26216