Theorem ruclem25 4909

Description: Lemma for ruc 4924. At any index A, the value of G is less than the value of H.

Hypotheses

Ref	Expression
ruclem.0	⊢ F:ℕ–→ℝ
ruclem.1	⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
ruclem.2	⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
ruclem.3	⊢ G = (1st ∘ (DseqC))
ruclem.4	⊢ H = (2nd ∘ (DseqC))
ruclem18.a	⊢ A ∈ ℕ

Assertion

Ref	Expression
ruclem25	⊢ (G ‘A) < (H ‘A)

Distinct variable group(s):   x,y,z   z,F

Proof of Theorem ruclem25

Step	Hyp	Ref	Expression
1		ruclem18.a	. 2 ⊢ A ∈ ℕ
2		fveq2 2832	. . . 4 ⊢ (w = 1 → (G ‘w) = (G ‘1))
3		fveq2 2832	. . . 4 ⊢ (w = 1 → (H ‘w) = (H ‘1))
4	2, 3	breq12d 2073	. . 3 ⊢ (w = 1 → ((G ‘w) < (H ‘w) ↔ (G ‘1) < (H ‘1)))
5		fveq2 2832	. . . 4 ⊢ (w = v → (G ‘w) = (G ‘v))
6		fveq2 2832	. . . 4 ⊢ (w = v → (H ‘w) = (H ‘v))
7	5, 6	breq12d 2073	. . 3 ⊢ (w = v → ((G ‘w) < (H ‘w) ↔ (G ‘v) < (H ‘v)))
8		fveq2 2832	. . . 4 ⊢ (w = (v + 1) → (G ‘w) = (G ‘(v + 1)))
9		fveq2 2832	. . . 4 ⊢ (w = (v + 1) → (H ‘w) = (H ‘(v + 1)))
10	8, 9	breq12d 2073	. . 3 ⊢ (w = (v + 1) → ((G ‘w) < (H ‘w) ↔ (G ‘(v + 1)) < (H ‘(v + 1))))
11		fveq2 2832	. . . 4 ⊢ (w = A → (G ‘w) = (G ‘A))
12		fveq2 2832	. . . 4 ⊢ (w = A → (H ‘w) = (H ‘A))
13	11, 12	breq12d 2073	. . 3 ⊢ (w = A → ((G ‘w) < (H ‘w) ↔ (G ‘A) < (H ‘A)))
14		ax1re 4064	. . . . . . 7 ⊢ 1 ∈ ℝ
15	14	ltplus1 4384	. . . . . 6 ⊢ 1 < (1 + 1)
16		df-2 4462	. . . . . 6 ⊢ 2 = (1 + 1)
17	15, 16	breqtrr 2082	. . . . 5 ⊢ 1 < 2
18		2re 4470	. . . . . 6 ⊢ 2 ∈ ℝ
19		ruclem.0	. . . . . . 7 ⊢ F:ℕ–→ℝ
20		1nn 4432	. . . . . . 7 ⊢ 1 ∈ ℕ
21		ffvrn 2890	. . . . . . 7 ⊢ ((F:ℕ–→ℝ ∧ 1 ∈ ℕ) → (F ‘1) ∈ ℝ)
22	19, 20, 21	mp2an 520	. . . . . 6 ⊢ (F ‘1) ∈ ℝ
23	14, 18, 22	ltadd2 4312	. . . . 5 ⊢ (1 < 2 ↔ ((F ‘1) + 1) < ((F ‘1) + 2))
24	17, 23	mpbi 164	. . . 4 ⊢ ((F ‘1) + 1) < ((F ‘1) + 2)
25		ruclem.1	. . . . 5 ⊢ C = ({⟨1, ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩⟩} ∪ (F ↾ (ℕ ∖ {1})))
26		ruclem.2	. . . . 5 ⊢ D = {⟨⟨x, y⟩, z⟩∣((x ∈ (ℝ × ℝ) ∧ y ∈ ℝ) ∧ z = if(((1st ‘x) < y ∧ y < (2nd ‘x)), ⟨(((2 · y) + (2nd ‘x)) / 3), ((y + (2 · (2nd ‘x))) / 3)⟩, ⟨(((2 · (1st ‘x)) + (2nd ‘x)) / 3), (((1st ‘x) + (2 · (2nd ‘x))) / 3)⟩))}
27		ruclem.3	. . . . 5 ⊢ G = (1st ∘ (DseqC))
28		ruclem.4	. . . . 5 ⊢ H = (2nd ∘ (DseqC))
29	19, 25, 26, 27, 28	ruclem16 4900	. . . 4 ⊢ (G ‘1) = ((F ‘1) + 1)
30	19, 25, 26, 27, 28	ruclem14 4898	. . . . . 6 ⊢ ((DseqC) ‘1) = ⟨((F ‘1) + 1), ((F ‘1) + 2)⟩
31	30	fveq2i 2835	. . . . 5 ⊢ (2nd ‘((DseqC) ‘1)) = (2nd ‘⟨((F ‘1) + 1), ((F ‘1) + 2)⟩)
32	26	ruclem9 4893	. . . . . 6 ⊢ D ∈ V
33	19, 25	ruclem5 4889	. . . . . 6 ⊢ C ∈ V
34	20, 32, 33, 28	ruclem11 4895	. . . . 5 ⊢ (2nd ‘((DseqC) ‘1)) = (H ‘1)
35		oprex 3018	. . . . . 6 ⊢ ((F ‘1) + 1) ∈ V
36		oprex 3018	. . . . . 6 ⊢ ((F ‘1) + 2) ∈ V
37	35, 36	op2nd 3092	. . . . 5 ⊢ (2nd ‘⟨((F ‘1) + 1), ((F ‘1) + 2)⟩) = ((F ‘1) + 2)
38	31, 34, 37	3eqtr3 1124	. . . 4 ⊢ (H ‘1) = ((F ‘1) + 2)
39	24, 29, 38	3brtr4 2085	. . 3 ⊢ (G ‘1) < (H ‘1)
40		fveq2 2832	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → (G ‘v) = (G ‘if(v ∈ ℕ, v, 1)))
41		fveq2 2832	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → (H ‘v) = (H ‘if(v ∈ ℕ, v, 1)))
42	40, 41	breq12d 2073	. . . . 5 ⊢ (v = if(v ∈ ℕ, v, 1) → ((G ‘v) < (H ‘v) ↔ (G ‘if(v ∈ ℕ, v, 1)) < (H ‘if(v ∈ ℕ, v, 1))))
43		opreq1 3006	. . . . . . 7 ⊢ (v = if(v ∈ ℕ, v, 1) → (v + 1) = (if(v ∈ ℕ, v, 1) + 1))
44	43	fveq2d 2836	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → (G ‘(v + 1)) = (G ‘(if(v ∈ ℕ, v, 1) + 1)))
45	43	fveq2d 2836	. . . . . 6 ⊢ (v = if(v ∈ ℕ, v, 1) → (H ‘(v + 1)) = (H ‘(if(v ∈ ℕ, v, 1) + 1)))
46	44, 45	breq12d 2073	. . . . 5 ⊢ (v = if(v ∈ ℕ, v, 1) → ((G ‘(v + 1)) < (H ‘(v + 1)) ↔ (G ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1))))
47	42, 46	imbi12d 474	. . . 4 ⊢ (v = if(v ∈ ℕ, v, 1) → (((G ‘v) < (H ‘v) → (G ‘(v + 1)) < (H ‘(v + 1))) ↔ ((G ‘if(v ∈ ℕ, v, 1)) < (H ‘if(v ∈ ℕ, v, 1)) → (G ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1)))))
48	20	elimel 1793	. . . . 5 ⊢ if(v ∈ ℕ, v, 1) ∈ ℕ
49	19, 25, 26, 27, 28, 48	ruclem24 4908	. . . 4 ⊢ ((G ‘if(v ∈ ℕ, v, 1)) < (H ‘if(v ∈ ℕ, v, 1)) → (G ‘(if(v ∈ ℕ, v, 1) + 1)) < (H ‘(if(v ∈ ℕ, v, 1) + 1)))
50	47, 49	dedth 1784	. . 3 ⊢ (v ∈ ℕ → ((G ‘v) < (H ‘v) → (G ‘(v + 1)) < (H ‘(v + 1))))
51	4, 7, 10, 13, 39, 50	nnind 4434	. 2 ⊢ (A ∈ ℕ → (G ‘A) < (H ‘A))
52	1, 51	ax-mp 6	1 ⊢ (G ‘A) < (H ‘A)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092   ∖ cdif 1484   ∪ cun 1485  ifcif 1776  {csn 1808  ⟨cop 1810   class class class wbr 2054   × cxp 2408   ↾ cres 2412   ∘ ccom 2414  –→wf 2418   ‘cfv 2422  (class class class)co 3001  {copab2 3002  1^st c1st 3085  2^nd c2nd 3086  ℝcr 4027  1c1 4029   + caddc 4031   · cmulc 4032   < clt 4033   / cdiv 4091  ℕcn 4093  2c2 4454  3c3 4455  seqcseq 4660

This theorem is referenced by:  ruclem26 4910  ruclem27 4911  ruclem32 4916

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-n0 4535  df-z 4564  df-seq 4661

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