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Theorem qabvle 27471

Description: By using induction on 𝑁, we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
qabsabv.a	⊢ 𝐴 = (AbsVal‘𝑄)

**Assertion**
Ref	Expression
qabvle	⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)

Proof of Theorem qabvle Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6891	. . . . 5 ⊢ (𝑘 = 0 → (𝐹‘𝑘) = (𝐹‘0))
2		id 22	. . . . 5 ⊢ (𝑘 = 0 → 𝑘 = 0)
3	1, 2	breq12d 5161	. . . 4 ⊢ (𝑘 = 0 → ((𝐹‘𝑘) ≤ 𝑘 ↔ (𝐹‘0) ≤ 0))
4	3	imbi2d 340	. . 3 ⊢ (𝑘 = 0 → ((𝐹 ∈ 𝐴 → (𝐹‘𝑘) ≤ 𝑘) ↔ (𝐹 ∈ 𝐴 → (𝐹‘0) ≤ 0)))
5		fveq2 6891	. . . . 5 ⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛))
6		id 22	. . . . 5 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
7	5, 6	breq12d 5161	. . . 4 ⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ≤ 𝑘 ↔ (𝐹‘𝑛) ≤ 𝑛))
8	7	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑛 → ((𝐹 ∈ 𝐴 → (𝐹‘𝑘) ≤ 𝑘) ↔ (𝐹 ∈ 𝐴 → (𝐹‘𝑛) ≤ 𝑛)))
9		fveq2 6891	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1)))
10		id 22	. . . . 5 ⊢ (𝑘 = (𝑛 + 1) → 𝑘 = (𝑛 + 1))
11	9, 10	breq12d 5161	. . . 4 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ≤ 𝑘 ↔ (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1)))
12	11	imbi2d 340	. . 3 ⊢ (𝑘 = (𝑛 + 1) → ((𝐹 ∈ 𝐴 → (𝐹‘𝑘) ≤ 𝑘) ↔ (𝐹 ∈ 𝐴 → (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1))))
13		fveq2 6891	. . . . 5 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁))
14		id 22	. . . . 5 ⊢ (𝑘 = 𝑁 → 𝑘 = 𝑁)
15	13, 14	breq12d 5161	. . . 4 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ≤ 𝑘 ↔ (𝐹‘𝑁) ≤ 𝑁))
16	15	imbi2d 340	. . 3 ⊢ (𝑘 = 𝑁 → ((𝐹 ∈ 𝐴 → (𝐹‘𝑘) ≤ 𝑘) ↔ (𝐹 ∈ 𝐴 → (𝐹‘𝑁) ≤ 𝑁)))
17		qabsabv.a	. . . . 5 ⊢ 𝐴 = (AbsVal‘𝑄)
18		qrng.q	. . . . . 6 ⊢ 𝑄 = (ℂ_fld ↾_s ℚ)
19	18	qrng0 27467	. . . . 5 ⊢ 0 = (0_g‘𝑄)
20	17, 19	abv0 20670	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘0) = 0)
21		0le0 12320	. . . 4 ⊢ 0 ≤ 0
22	20, 21	eqbrtrdi 5187	. . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘0) ≤ 0)
23		nn0p1nn 12518	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ₀ → (𝑛 + 1) ∈ ℕ)
24	23	ad2antrl 725	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝑛 + 1) ∈ ℕ)
25		nnq 12953	. . . . . . . . 9 ⊢ ((𝑛 + 1) ∈ ℕ → (𝑛 + 1) ∈ ℚ)
26	24, 25	syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝑛 + 1) ∈ ℚ)
27	18	qrngbas 27465	. . . . . . . . 9 ⊢ ℚ = (Base‘𝑄)
28	17, 27	abvcl 20663	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 + 1) ∈ ℚ) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
29	26, 28	syldan 590	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘(𝑛 + 1)) ∈ ℝ)
30		nn0z 12590	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ₀ → 𝑛 ∈ ℤ)
31	30	ad2antrl 725	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 𝑛 ∈ ℤ)
32		zq 12945	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℚ)
33	31, 32	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 𝑛 ∈ ℚ)
34	17, 27	abvcl 20663	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ) → (𝐹‘𝑛) ∈ ℝ)
35	33, 34	syldan 590	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘𝑛) ∈ ℝ)
36		peano2re 11394	. . . . . . . 8 ⊢ ((𝐹‘𝑛) ∈ ℝ → ((𝐹‘𝑛) + 1) ∈ ℝ)
37	35, 36	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → ((𝐹‘𝑛) + 1) ∈ ℝ)
38	31	zred 12673	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 𝑛 ∈ ℝ)
39		peano2re 11394	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ → (𝑛 + 1) ∈ ℝ)
40	38, 39	syl 17	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝑛 + 1) ∈ ℝ)
41		simpl 482	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 𝐹 ∈ 𝐴)
42		1z 12599	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
43		zq 12945	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
44	42, 43	mp1i 13	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 1 ∈ ℚ)
45		qex 12952	. . . . . . . . . . 11 ⊢ ℚ ∈ V
46		cnfldadd 21238	. . . . . . . . . . . 12 ⊢ + = (+_g‘ℂ_fld)
47	18, 46	ressplusg 17242	. . . . . . . . . . 11 ⊢ (ℚ ∈ V → + = (+_g‘𝑄))
48	45, 47	ax-mp 5	. . . . . . . . . 10 ⊢ + = (+_g‘𝑄)
49	17, 27, 48	abvtri 20669	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐹‘(𝑛 + 1)) ≤ ((𝐹‘𝑛) + (𝐹‘1)))
50	41, 33, 44, 49	syl3anc 1370	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘(𝑛 + 1)) ≤ ((𝐹‘𝑛) + (𝐹‘1)))
51		ax-1ne0 11185	. . . . . . . . . . 11 ⊢ 1 ≠ 0
52	18	qrng1 27468	. . . . . . . . . . . 12 ⊢ 1 = (1_r‘𝑄)
53	17, 52, 19	abv1z 20671	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ 𝐴 ∧ 1 ≠ 0) → (𝐹‘1) = 1)
54	51, 53	mpan2 688	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘1) = 1)
55	54	adantr 480	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘1) = 1)
56	55	oveq2d 7428	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → ((𝐹‘𝑛) + (𝐹‘1)) = ((𝐹‘𝑛) + 1))
57	50, 56	breqtrd 5174	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘(𝑛 + 1)) ≤ ((𝐹‘𝑛) + 1))
58		1red 11222	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → 1 ∈ ℝ)
59		simprr 770	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘𝑛) ≤ 𝑛)
60	35, 38, 58, 59	leadd1dd 11835	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → ((𝐹‘𝑛) + 1) ≤ (𝑛 + 1))
61	29, 37, 40, 57, 60	letrd 11378	. . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑛 ∈ ℕ₀ ∧ (𝐹‘𝑛) ≤ 𝑛)) → (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1))
62	61	expr 456	. . . . 5 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑛 ∈ ℕ₀) → ((𝐹‘𝑛) ≤ 𝑛 → (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1)))
63	62	expcom 413	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (𝐹 ∈ 𝐴 → ((𝐹‘𝑛) ≤ 𝑛 → (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1))))
64	63	a2d 29	. . 3 ⊢ (𝑛 ∈ ℕ₀ → ((𝐹 ∈ 𝐴 → (𝐹‘𝑛) ≤ 𝑛) → (𝐹 ∈ 𝐴 → (𝐹‘(𝑛 + 1)) ≤ (𝑛 + 1))))
65	4, 8, 12, 16, 22, 64	nn0ind 12664	. 2 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∈ 𝐴 → (𝐹‘𝑁) ≤ 𝑁))
66	65	impcom 407	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ₀) → (𝐹‘𝑁) ≤ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 Vcvv 3473 class class class wbr 5148 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 + caddc 11119 ≤ cle 11256 ℕcn 12219 ℕ₀cn0 12479 ℤcz 12565 ℚcq 12939 ↾_s cress 17180 +_gcplusg 17204 AbsValcabv 20655 ℂ_fldccnfld 21233

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-addf 11195 ax-mulf 11196

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-q 12940 df-ico 13337 df-fz 13492 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 df-minusg 18865 df-subg 19046 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-ring 20136 df-cring 20137 df-oppr 20232 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-subrng 20442 df-subrg 20467 df-drng 20585 df-abv 20656 df-cnfld 21234

This theorem is referenced by: ostth2lem2 27480 ostth2 27483

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