Theorem nninfdclemlt 13286

Description: Lemma for nninfdc 13288. The function from nninfdclemf 13284 is strictly monotonic. (Contributed by Jim Kingdon, 24-Sep-2024.)

Hypotheses

Ref	Expression
nninfdclemf.a	|- ( ph -> A C_ NN )
nninfdclemf.dc	|- ( ph -> A. x e. NN DECID x e. A )
nninfdclemf.nb	|- ( ph -> A. m e. NN E. n e. A m < n )
nninfdclemf.j	|- ( ph -> ( J e. A /\ 1 < J ) )
nninfdclemf.f	|- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )
nninfdclemlt.u	|- ( ph -> U e. NN )
nninfdclemlt.v	|- ( ph -> V e. NN )
nninfdclemlt.lt	|- ( ph -> U < V )

Assertion

Ref	Expression
nninfdclemlt	|- ( ph -> ( F ` U ) < ( F ` V ) )

Distinct variable groups:   A, m, n   x, A   y, A, z   m, F, n   x, F   y, F, z   i, J   U, i   U, m, n   x, U   y, U, z   y, J, z

Allowed substitution hints:   ph( x, y, z, i, m, n)   A( i)   F( i)   J( x, m, n)   V( x, y, z, i, m, n)

Proof of Theorem nninfdclemlt

Dummy variables k w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfdclemlt.u	. . . . . 6 |- ( ph -> U e. NN )
2	1	peano2nnd 9269	. . . . 5 |- ( ph -> ( U + 1 ) e. NN )
3	2	nnzd 9717	. . . 4 |- ( ph -> ( U + 1 ) e. ZZ )
4		nninfdclemlt.v	. . . . 5 |- ( ph -> V e. NN )
5	4	nnzd 9717	. . . 4 |- ( ph -> V e. ZZ )
6		nninfdclemlt.lt	. . . . 5 |- ( ph -> U < V )
7		nnltp1le 9655	. . . . . 6 |- ( ( U e. NN /\ V e. NN ) -> ( U < V <-> ( U + 1 ) <_ V ) )
8	1, 4, 7	syl2anc 411	. . . . 5 |- ( ph -> ( U < V <-> ( U + 1 ) <_ V ) )
9	6, 8	mpbid 147	. . . 4 |- ( ph -> ( U + 1 ) <_ V )
10		eluz2 9877	. . . 4 |- ( V e. ( ZZ>= ` ( U + 1 ) ) <-> ( ( U + 1 ) e. ZZ /\ V e. ZZ /\ ( U + 1 ) <_ V ) )
11	3, 5, 9, 10	syl3anbrc 1208	. . 3 |- ( ph -> V e. ( ZZ>= ` ( U + 1 ) ) )
12		eluzfz2 10386	. . 3 |- ( V e. ( ZZ>= ` ( U + 1 ) ) -> V e. ( ( U + 1 ) ... V ) )
13	11, 12	syl 14	. 2 |- ( ph -> V e. ( ( U + 1 ) ... V ) )
14		fveq2 5675	. . . . 5 |- ( w = ( U + 1 ) -> ( F ` w ) = ( F ` ( U + 1 ) ) )
15	14	breq2d 4126	. . . 4 |- ( w = ( U + 1 ) -> ( ( F ` U ) < ( F ` w ) <-> ( F ` U ) < ( F ` ( U + 1 ) ) ) )
16	15	imbi2d 230	. . 3 |- ( w = ( U + 1 ) -> ( ( ph -> ( F ` U ) < ( F ` w ) ) <-> ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) ) ) )
17		fveq2 5675	. . . . 5 |- ( w = k -> ( F ` w ) = ( F ` k ) )
18	17	breq2d 4126	. . . 4 |- ( w = k -> ( ( F ` U ) < ( F ` w ) <-> ( F ` U ) < ( F ` k ) ) )
19	18	imbi2d 230	. . 3 |- ( w = k -> ( ( ph -> ( F ` U ) < ( F ` w ) ) <-> ( ph -> ( F ` U ) < ( F ` k ) ) ) )
20		fveq2 5675	. . . . 5 |- ( w = ( k + 1 ) -> ( F ` w ) = ( F ` ( k + 1 ) ) )
21	20	breq2d 4126	. . . 4 |- ( w = ( k + 1 ) -> ( ( F ` U ) < ( F ` w ) <-> ( F ` U ) < ( F ` ( k + 1 ) ) ) )
22	21	imbi2d 230	. . 3 |- ( w = ( k + 1 ) -> ( ( ph -> ( F ` U ) < ( F ` w ) ) <-> ( ph -> ( F ` U ) < ( F ` ( k + 1 ) ) ) ) )
23		fveq2 5675	. . . . 5 |- ( w = V -> ( F ` w ) = ( F ` V ) )
24	23	breq2d 4126	. . . 4 |- ( w = V -> ( ( F ` U ) < ( F ` w ) <-> ( F ` U ) < ( F ` V ) ) )
25	24	imbi2d 230	. . 3 |- ( w = V -> ( ( ph -> ( F ` U ) < ( F ` w ) ) <-> ( ph -> ( F ` U ) < ( F ` V ) ) ) )
26		nninfdclemf.a	. . . . 5 |- ( ph -> A C_ NN )
27		nninfdclemf.dc	. . . . 5 |- ( ph -> A. x e. NN DECID x e. A )
28		nninfdclemf.nb	. . . . 5 |- ( ph -> A. m e. NN E. n e. A m < n )
29		nninfdclemf.j	. . . . 5 |- ( ph -> ( J e. A /\ 1 < J ) )
30		nninfdclemf.f	. . . . 5 |- F = seq 1 ( ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) , ( i e. NN |-> J ) )
31	26, 27, 28, 29, 30, 1	nninfdclemp1 13285	. . . 4 |- ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) )
32	31	a1i 9	. . 3 |- ( V e. ( ZZ>= ` ( U + 1 ) ) -> ( ph -> ( F ` U ) < ( F ` ( U + 1 ) ) ) )
33	26	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A C_ NN )
34	26, 27, 28, 29, 30	nninfdclemf 13284	. . . . . . . . . . 11 |- ( ph -> F : NN --> A )
35	34	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> F : NN --> A )
36	1	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> U e. NN )
37	35, 36	ffvelcdmd 5818	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. A )
38	33, 37	sseldd 3243	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. NN )
39	38	nnred 9267	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. RR )
40		elfzoelz 10503	. . . . . . . . . . . 12 |- ( k e. ( ( U + 1 )..^ V ) -> k e. ZZ )
41	40	ad2antlr 489	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. ZZ )
42		1red 8305	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 e. RR )
43	2	nnred 9267	. . . . . . . . . . . . 13 |- ( ph -> ( U + 1 ) e. RR )
44	43	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( U + 1 ) e. RR )
45	41	zred 9718	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. RR )
46	2	nnge1d 9297	. . . . . . . . . . . . 13 |- ( ph -> 1 <_ ( U + 1 ) )
47	46	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 <_ ( U + 1 ) )
48		elfzole1 10512	. . . . . . . . . . . . 13 |- ( k e. ( ( U + 1 )..^ V ) -> ( U + 1 ) <_ k )
49	48	ad2antlr 489	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( U + 1 ) <_ k )
50	42, 44, 45, 47, 49	letrd 8413	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 <_ k )
51		elnnz1 9617	. . . . . . . . . . 11 |- ( k e. NN <-> ( k e. ZZ /\ 1 <_ k ) )
52	41, 50, 51	sylanbrc 417	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. NN )
53	35, 52	ffvelcdmd 5818	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. A )
54	33, 53	sseldd 3243	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. NN )
55	54	nnred 9267	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. RR )
56	52	peano2nnd 9269	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( k + 1 ) e. NN )
57	35, 56	ffvelcdmd 5818	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. A )
58	33, 57	sseldd 3243	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. NN )
59	58	nnred 9267	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. RR )
60		simpr 110	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) < ( F ` k ) )
61	27	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A. x e. NN DECID x e. A )
62	28	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A. m e. NN E. n e. A m < n )
63	29	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( J e. A /\ 1 < J ) )
64	33, 61, 62, 63, 30, 52	nninfdclemp1 13285	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) )
65	39, 55, 59, 60, 64	lttrd 8415	. . . . . 6 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) < ( F ` ( k + 1 ) ) )
66	65	ex 115	. . . . 5 |- ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) -> ( ( F ` U ) < ( F ` k ) -> ( F ` U ) < ( F ` ( k + 1 ) ) ) )
67	66	expcom 116	. . . 4 |- ( k e. ( ( U + 1 )..^ V ) -> ( ph -> ( ( F ` U ) < ( F ` k ) -> ( F ` U ) < ( F ` ( k + 1 ) ) ) ) )
68	67	a2d 26	. . 3 |- ( k e. ( ( U + 1 )..^ V ) -> ( ( ph -> ( F ` U ) < ( F ` k ) ) -> ( ph -> ( F ` U ) < ( F ` ( k + 1 ) ) ) ) )
69	16, 19, 22, 25, 32, 68	fzind2 10607	. 2 |- ( V e. ( ( U + 1 ) ... V ) -> ( ph -> ( F ` U ) < ( F ` V ) ) )
70	13, 69	mpcom 36	1 |- ( ph -> ( F ` U ) < ( F ` V ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   i^i cin 3213   C_ wss 3214   class class class wbr 4114   |-> cmpt 4176   -->wf 5353   ` cfv 5357  (class class class)co 6058   e. cmpo 6060  infcinf 7287   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   <_ cle 8325   NNcn 9254   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ..^cfzo 10498   seqcseq 10833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-sup 7288  df-inf 7289  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834

This theorem is referenced by:  nninfdclemf1  13287