Theorem nninfdclemlt 12937

Description: Lemma for nninfdc 12939. The function from nninfdclemf 12935 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 9086	. . . . 5 |- ( ph -> ( U + 1 ) e. NN )
3	2	nnzd 9529	. . . 4 |- ( ph -> ( U + 1 ) e. ZZ )
4		nninfdclemlt.v	. . . . 5 |- ( ph -> V e. NN )
5	4	nnzd 9529	. . . 4 |- ( ph -> V e. ZZ )
6		nninfdclemlt.lt	. . . . 5 |- ( ph -> U < V )
7		nnltp1le 9468	. . . . . 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 9689	. . . 4 |- ( V e. ( ZZ>= ` ( U + 1 ) ) <-> ( ( U + 1 ) e. ZZ /\ V e. ZZ /\ ( U + 1 ) <_ V ) )
11	3, 5, 9, 10	syl3anbrc 1184	. . 3 |- ( ph -> V e. ( ZZ>= ` ( U + 1 ) ) )
12		eluzfz2 10189	. . 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 5599	. . . . 5 |- ( w = ( U + 1 ) -> ( F ` w ) = ( F ` ( U + 1 ) ) )
15	14	breq2d 4071	. . . 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 5599	. . . . 5 |- ( w = k -> ( F ` w ) = ( F ` k ) )
18	17	breq2d 4071	. . . 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 5599	. . . . 5 |- ( w = ( k + 1 ) -> ( F ` w ) = ( F ` ( k + 1 ) ) )
21	20	breq2d 4071	. . . 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 5599	. . . . 5 |- ( w = V -> ( F ` w ) = ( F ` V ) )
24	23	breq2d 4071	. . . 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 12936	. . . 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 12935	. . . . . . . . . . 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 5739	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. A )
38	33, 37	sseldd 3202	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. NN )
39	38	nnred 9084	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` U ) e. RR )
40		elfzoelz 10304	. . . . . . . . . . . 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 8122	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 e. RR )
43	2	nnred 9084	. . . . . . . . . . . . 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 9530	. . . . . . . . . . . 12 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> k e. RR )
46	2	nnge1d 9114	. . . . . . . . . . . . 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 10313	. . . . . . . . . . . . 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 8231	. . . . . . . . . . 11 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> 1 <_ k )
51		elnnz1 9430	. . . . . . . . . . 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 5739	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. A )
54	33, 53	sseldd 3202	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. NN )
55	54	nnred 9084	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) e. RR )
56	52	peano2nnd 9086	. . . . . . . . . 10 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( k + 1 ) e. NN )
57	35, 56	ffvelcdmd 5739	. . . . . . . . 9 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. A )
58	33, 57	sseldd 3202	. . . . . . . 8 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` ( k + 1 ) ) e. NN )
59	58	nnred 9084	. . . . . . 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 12936	. . . . . . 7 |- ( ( ( ph /\ k e. ( ( U + 1 )..^ V ) ) /\ ( F ` U ) < ( F ` k ) ) -> ( F ` k ) < ( F ` ( k + 1 ) ) )
65	39, 55, 59, 60, 64	lttrd 8233	. . . . . 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 10405	. 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 836   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   i^i cin 3173   C_ wss 3174   class class class wbr 4059   |-> cmpt 4121   -->wf 5286   ` cfv 5290  (class class class)co 5967   e. cmpo 5969  infcinf 7111   RRcr 7959   1c1 7961   + caddc 7963   < clt 8142   <_ cle 8143   NNcn 9071   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165  ..^cfzo 10299   seqcseq 10629

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-seqfrec 10630

This theorem is referenced by:  nninfdclemf1  12938