Theorem bcthlem2 9278

Description: Lemma for bcth 9310. For any m, we can always find a greater n meeting the convergence criterion.

Assertion

Ref	Expression
bcthlem2	|- (((m e. NN /\ j e. NN) /\ A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))

Distinct variable groups:   k,n,D   S,k,n   g,k,n   j,k,n   k,m,n   k,q,n

Proof of Theorem bcthlem2

Step	Hyp	Ref	Expression
1		breq2 3342	. . . . . . 7 |- (k = j -> (j <_ k <-> j <_ j))
2		fveq2 4681	. . . . . . . . 9 |- (k = j -> ((1st o. g)` k) = ((1st o. g)` j))
3	2	opreq1d 4897	. . . . . . . 8 |- (k = j -> (((1st o. g)` k)Dq) = (((1st o. g)` j)Dq))
4	3	breq1d 3348	. . . . . . 7 |- (k = j -> ((((1st o. g)` k)Dq) < S <-> (((1st o. g)` j)Dq) < S))
5	1, 4	imbi12d 688	. . . . . 6 |- (k = j -> ((j <_ k -> (((1st o. g)` k)Dq) < S) <-> (j <_ j -> (((1st o. g)` j)Dq) < S)))
6	5	rcla4v 2376	. . . . 5 |- (j e. NN -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> (j <_ j -> (((1st o. g)` j)Dq) < S)))
7	6	ad2antlr 441	. . . 4 |- (((m e. NN /\ j e. NN) /\ m < j) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> (j <_ j -> (((1st o. g)` j)Dq) < S)))
8		nnre 7112	. . . . . . 7 |- (j e. NN -> j e. RR)
9		leid 6701	. . . . . . 7 |- (j e. RR -> j <_ j)
10	8, 9	syl 12	. . . . . 6 |- (j e. NN -> j <_ j)
11	10	ad2antlr 441	. . . . 5 |- (((m e. NN /\ j e. NN) /\ m < j) -> j <_ j)
12		pm2.27 76	. . . . 5 |- (j <_ j -> ((j <_ j -> (((1st o. g)` j)Dq) < S) -> (((1st o. g)` j)Dq) < S))
13	11, 12	syl 12	. . . 4 |- (((m e. NN /\ j e. NN) /\ m < j) -> ((j <_ j -> (((1st o. g)` j)Dq) < S) -> (((1st o. g)` j)Dq) < S))
14		simpllr 453	. . . . . 6 |- ((((m e. NN /\ j e. NN) /\ m < j) /\ (((1st o. g)` j)Dq) < S) -> j e. NN)
15		id 73	. . . . . . 7 |- ((m < j /\ (((1st o. g)` j)Dq) < S) -> (m < j /\ (((1st o. g)` j)Dq) < S))
16	15	adantll 428	. . . . . 6 |- ((((m e. NN /\ j e. NN) /\ m < j) /\ (((1st o. g)` j)Dq) < S) -> (m < j /\ (((1st o. g)` j)Dq) < S))
17		breq2 3342	. . . . . . . 8 |- (n = j -> (m < n <-> m < j))
18		fveq2 4681	. . . . . . . . . 10 |- (n = j -> ((1st o. g)` n) = ((1st o. g)` j))
19	18	opreq1d 4897	. . . . . . . . 9 |- (n = j -> (((1st o. g)` n)Dq) = (((1st o. g)` j)Dq))
20	19	breq1d 3348	. . . . . . . 8 |- (n = j -> ((((1st o. g)` n)Dq) < S <-> (((1st o. g)` j)Dq) < S))
21	17, 20	anbi12d 690	. . . . . . 7 |- (n = j -> ((m < n /\ (((1st o. g)` n)Dq) < S) <-> (m < j /\ (((1st o. g)` j)Dq) < S)))
22	21	rcla4ev 2381	. . . . . 6 |- ((j e. NN /\ (m < j /\ (((1st o. g)` j)Dq) < S)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))
23	14, 16, 22	syl11anc 524	. . . . 5 |- ((((m e. NN /\ j e. NN) /\ m < j) /\ (((1st o. g)` j)Dq) < S) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))
24	23	ex 402	. . . 4 |- (((m e. NN /\ j e. NN) /\ m < j) -> ((((1st o. g)` j)Dq) < S -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S)))
25	7, 13, 24	3syld 31	. . 3 |- (((m e. NN /\ j e. NN) /\ m < j) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S)))
26		peano2nn 7118	. . . . . 6 |- (m e. NN -> (m + 1) e. NN)
27		breq2 3342	. . . . . . . 8 |- (k = (m + 1) -> (j <_ k <-> j <_ (m + 1)))
28		fveq2 4681	. . . . . . . . . 10 |- (k = (m + 1) -> ((1st o. g)` k) = ((1st o. g)` (m + 1)))
29	28	opreq1d 4897	. . . . . . . . 9 |- (k = (m + 1) -> (((1st o. g)` k)Dq) = (((1st o. g)` (m + 1))Dq))
30	29	breq1d 3348	. . . . . . . 8 |- (k = (m + 1) -> ((((1st o. g)` k)Dq) < S <-> (((1st o. g)` (m + 1))Dq) < S))
31	27, 30	imbi12d 688	. . . . . . 7 |- (k = (m + 1) -> ((j <_ k -> (((1st o. g)` k)Dq) < S) <-> (j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S)))
32	31	rcla4v 2376	. . . . . 6 |- ((m + 1) e. NN -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> (j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S)))
33	26, 32	syl 12	. . . . 5 |- (m e. NN -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> (j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S)))
34	33	ad2antrr 440	. . . 4 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> (j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S)))
35		nnre 7112	. . . . . . . . 9 |- (m e. NN -> m e. RR)
36		lep1 6990	. . . . . . . . 9 |- (m e. RR -> m <_ (m + 1))
37	35, 36	syl 12	. . . . . . . 8 |- (m e. NN -> m <_ (m + 1))
38	37	adantr 425	. . . . . . 7 |- ((m e. NN /\ j e. NN) -> m <_ (m + 1))
39	8	adantl 424	. . . . . . . 8 |- ((m e. NN /\ j e. NN) -> j e. RR)
40	35	adantr 425	. . . . . . . 8 |- ((m e. NN /\ j e. NN) -> m e. RR)
41		peano2re 6599	. . . . . . . . . 10 |- (m e. RR -> (m + 1) e. RR)
42	35, 41	syl 12	. . . . . . . . 9 |- (m e. NN -> (m + 1) e. RR)
43	42	adantr 425	. . . . . . . 8 |- ((m e. NN /\ j e. NN) -> (m + 1) e. RR)
44		letr 6695	. . . . . . . 8 |- ((j e. RR /\ m e. RR /\ (m + 1) e. RR) -> ((j <_ m /\ m <_ (m + 1)) -> j <_ (m + 1)))
45	39, 40, 43, 44	syl111anc 1100	. . . . . . 7 |- ((m e. NN /\ j e. NN) -> ((j <_ m /\ m <_ (m + 1)) -> j <_ (m + 1)))
46	38, 45	mpan2d 766	. . . . . 6 |- ((m e. NN /\ j e. NN) -> (j <_ m -> j <_ (m + 1)))
47	46	imp 377	. . . . 5 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> j <_ (m + 1))
48		pm2.27 76	. . . . 5 |- (j <_ (m + 1) -> ((j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S) -> (((1st o. g)` (m + 1))Dq) < S))
49	47, 48	syl 12	. . . 4 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> ((j <_ (m + 1) -> (((1st o. g)` (m + 1))Dq) < S) -> (((1st o. g)` (m + 1))Dq) < S))
50	26	ad2antrr 440	. . . . . . 7 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> (m + 1) e. NN)
51	50	adantr 425	. . . . . 6 |- ((((m e. NN /\ j e. NN) /\ j <_ m) /\ (((1st o. g)` (m + 1))Dq) < S) -> (m + 1) e. NN)
52		ltp1 6989	. . . . . . . . 9 |- (m e. RR -> m < (m + 1))
53	35, 52	syl 12	. . . . . . . 8 |- (m e. NN -> m < (m + 1))
54	53	ad2antrr 440	. . . . . . 7 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> m < (m + 1))
55	54	anim1i 361	. . . . . 6 |- ((((m e. NN /\ j e. NN) /\ j <_ m) /\ (((1st o. g)` (m + 1))Dq) < S) -> (m < (m + 1) /\ (((1st o. g)` (m + 1))Dq) < S))
56		breq2 3342	. . . . . . . 8 |- (n = (m + 1) -> (m < n <-> m < (m + 1)))
57		fveq2 4681	. . . . . . . . . 10 |- (n = (m + 1) -> ((1st o. g)` n) = ((1st o. g)` (m + 1)))
58	57	opreq1d 4897	. . . . . . . . 9 |- (n = (m + 1) -> (((1st o. g)` n)Dq) = (((1st o. g)` (m + 1))Dq))
59	58	breq1d 3348	. . . . . . . 8 |- (n = (m + 1) -> ((((1st o. g)` n)Dq) < S <-> (((1st o. g)` (m + 1))Dq) < S))
60	56, 59	anbi12d 690	. . . . . . 7 |- (n = (m + 1) -> ((m < n /\ (((1st o. g)` n)Dq) < S) <-> (m < (m + 1) /\ (((1st o. g)` (m + 1))Dq) < S)))
61	60	rcla4ev 2381	. . . . . 6 |- (((m + 1) e. NN /\ (m < (m + 1) /\ (((1st o. g)` (m + 1))Dq) < S)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))
62	51, 55, 61	syl11anc 524	. . . . 5 |- ((((m e. NN /\ j e. NN) /\ j <_ m) /\ (((1st o. g)` (m + 1))Dq) < S) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))
63	62	ex 402	. . . 4 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> ((((1st o. g)` (m + 1))Dq) < S -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S)))
64	34, 49, 63	3syld 31	. . 3 |- (((m e. NN /\ j e. NN) /\ j <_ m) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S)))
65	25, 64, 40, 39	pm2.61ltlei 6705	. 2 |- ((m e. NN /\ j e. NN) -> (A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S)))
66	65	imp 377	1 |- (((m e. NN /\ j e. NN) /\ A.k e. NN (j <_ k -> (((1st o. g)` k)Dq) < S)) -> E.n e. NN (m < n /\ (((1st o. g)` n)Dq) < S))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338   o. ccom 3990  ` cfv 3998  (class class class)co 4884  1stc1st 5018  RRcr 6385  1c1 6387   + caddc 6389   <_ cle 6448  NNcn 6449   < clt 6653

This theorem is referenced by:  bcthlem13 9289

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108

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