Theorem nninfdclemlt 13320

Description: Lemma for nninfdc 13322. The function from nninfdclemf 13318 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 9298	. . . . 5 |- ( ph -> ( U + 1 ) e. NN )
3	2	nnzd 9746	. . . 4 |- ( ph -> ( U + 1 ) e. ZZ )
4		nninfdclemlt.v	. . . . 5 |- ( ph -> V e. NN )
5	4	nnzd 9746	. . . 4 |- ( ph -> V e. ZZ )
6		nninfdclemlt.lt	. . . . 5 |- ( ph -> U < V )
7		nnltp1le 9684	. . . . . 6 |- ( ( U e. NN /\ V e. NN ) -> ( U < V <-> ( U + 1 ) <_ V ) )
8	1, 4, 7	syl2anc 415	. . . . 5 |- ( ph -> ( U < V <-> ( U + 1 ) <_ V ) )
9	6, 8	mpbid 147	. . . 4 |- ( ph -> ( U + 1 ) <_ V )
10		eluz2 9906	. . . 4 |- ( V e. ( ZZ>= ` ( U + 1 ) ) <-> ( ( U + 1 ) e. ZZ /\ V e. ZZ /\ ( U + 1 ) <_ V ) )
11	3, 5, 9, 10	syl3anbrc 1212	. . 3 |- ( ph -> V e. ( ZZ>= ` ( U + 1 ) ) )
12		eluzfz2 10415	. . 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 5690	. . . . 5 |- ( w = ( U + 1 ) -> ( F ` w ) = ( F ` ( U + 1 ) ) )
15	14	breq2d 4137	. . . 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 5690	. . . . 5 |- ( w = k -> ( F ` w ) = ( F ` k ) )
18	17	breq2d 4137	. . . 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 5690	. . . . 5 |- ( w = ( k + 1 ) -> ( F ` w ) = ( F ` ( k + 1 ) ) )
21	20	breq2d 4137	. . . 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 5690	. . . . 5 |- ( w = V -> ( F ` w ) = ( F ` V ) )
24	23	breq2d 4137	. . . 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 13319	. . . 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 492	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A C_ NN )
34	26, 27, 28, 29, 30	nninfdclemf 13318	. . . . . . . . . . 11 |- ( ph -> F : NN --> A )
35	34	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> F : NN --> A )
36	1	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> U e. NN )
37	35, 36	ffvelcdmd 5835	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. A )
38	33, 37	sseldd 3249	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. NN )
39	38	nnred 9296	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. RR )
40		elfzoelz 10532	. . . . . . . . . . . 12 |- ( k e. ( ( U + 1 )..^ V ) -> k e. ZZ )
41	40	ad2antlr 493	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. ZZ )
42		1red 8331	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 e. RR )
43	2	nnred 9296	. . . . . . . . . . . . 13 |- ( ph -> ( U + 1 ) e. RR )
44	43	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( U + 1 ) e. RR )
45	41	zred 9747	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. RR )
46	2	nnge1d 9326	. . . . . . . . . . . . 13 |- ( ph -> 1 <_ ( U + 1 ) )
47	46	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 <_ ( U + 1 ) )
48		elfzole1 10541	. . . . . . . . . . . . 13 |- ( k e. ( ( U + 1 )..^ V ) -> ( U + 1 ) <_ k )
49	48	ad2antlr 493	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( U + 1 ) <_ k )
50	42, 44, 45, 47, 49	letrd 8440	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 <_ k )
51		elnnz1 9646	. . . . . . . . . . 11 |- ( k e. NN <-> ( k e. ZZ /\ 1 <_ k ) )
52	41, 50, 51	sylanbrc 421	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. NN )
53	35, 52	ffvelcdmd 5835	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. A )
54	33, 53	sseldd 3249	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. NN )
55	54	nnred 9296	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. RR )
56	52	peano2nnd 9298	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( k + 1 ) e. NN )
57	35, 56	ffvelcdmd 5835	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. A )
58	33, 57	sseldd 3249	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. NN )
59	58	nnred 9296	. . . . . . 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 492	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A. x e. NN DECID x e. A )
62	28	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> A. m e. NN E. n e. A m < n )
63	29	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( J e. A /\ 1 < J ) )
64	33, 61, 62, 63, 30, 52	nninfdclemp1 13319	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) )
65	39, 55, 59, 60, 64	lttrd 8442	. . . . . 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 10636	. 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 846   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   i^i cin 3219   C_ wss 3220   class class class wbr 4125   |-> cmpt 4187   -->wf 5368   ` cfv 5372  (class class class)co 6075   e. cmpo 6077  infcinf 7313   RRcr 8168   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   NNcn 9283   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390  ..^cfzo 10527   seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863

This theorem is referenced by:  nninfdclemf1  13321