Theorem nninfdclemcl 13199

Description: Lemma for nninfdc 13204. (Contributed by Jim Kingdon, 25-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 )
nninfdclemcl.p	|- ( ph -> P e. A )
nninfdclemcl.q	|- ( ph -> Q e. A )

Assertion

Ref	Expression
nninfdclemcl	|- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) e. A )

Distinct variable groups:   x, A   y, A, z   A, m, n   x, P   P, m, n   y, P, z   y, Q, z   m, n

Allowed substitution hints:   ph( x, y, z, m, n)   Q( x, m, n)

Proof of Theorem nninfdclemcl

Dummy variables r s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninfdclemf.a	. . . 4 |- ( ph -> A C_ NN )
2		nninfdclemcl.p	. . . 4 |- ( ph -> P e. A )
3	1, 2	sseldd 3239	. . 3 |- ( ph -> P e. NN )
4		nninfdclemcl.q	. . . 4 |- ( ph -> Q e. A )
5	1, 4	sseldd 3239	. . 3 |- ( ph -> Q e. NN )
6		inss1 3441	. . . . . 6 |- ( A i^i ( ZZ>= ` ( P + 1 ) ) ) C_ A
7	6, 1	sstrid 3249	. . . . 5 |- ( ph -> ( A i^i ( ZZ>= ` ( P + 1 ) ) ) C_ NN )
8		eleq1 2295	. . . . . . . . . . 11 |- ( x = s -> ( x e. A <-> s e. A ) )
9	8	dcbid 846	. . . . . . . . . 10 |- ( x = s -> (DECID x e. A <-> DECID s e. A ) )
10		nninfdclemf.dc	. . . . . . . . . . 11 |- ( ph -> A. x e. NN DECID x e. A )
11	10	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ s e. NN ) -> A. x e. NN DECID x e. A )
12		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ s e. NN ) -> s e. NN )
13	9, 11, 12	rspcdva 2926	. . . . . . . . 9 |- ( ( ph /\ s e. NN ) -> DECID s e. A )
14	3	adantr 276	. . . . . . . . . . . 12 |- ( ( ph /\ s e. NN ) -> P e. NN )
15	14	nnzd 9699	. . . . . . . . . . 11 |- ( ( ph /\ s e. NN ) -> P e. ZZ )
16	15	peano2zd 9703	. . . . . . . . . 10 |- ( ( ph /\ s e. NN ) -> ( P + 1 ) e. ZZ )
17	12	nnzd 9699	. . . . . . . . . 10 |- ( ( ph /\ s e. NN ) -> s e. ZZ )
18		eluzdc 9942	. . . . . . . . . 10 |- ( ( ( P + 1 ) e. ZZ /\ s e. ZZ ) -> DECID s e. ( ZZ>= ` ( P + 1 ) ) )
19	16, 17, 18	syl2anc 411	. . . . . . . . 9 |- ( ( ph /\ s e. NN ) -> DECID s e. ( ZZ>= ` ( P + 1 ) ) )
20	13, 19	dcand 941	. . . . . . . 8 |- ( ( ph /\ s e. NN ) -> DECID ( s e. A /\ s e. ( ZZ>= ` ( P + 1 ) ) ) )
21		elin 3402	. . . . . . . . 9 |- ( s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) <-> ( s e. A /\ s e. ( ZZ>= ` ( P + 1 ) ) ) )
22	21	dcbii 848	. . . . . . . 8 |- (DECID s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) <-> DECID ( s e. A /\ s e. ( ZZ>= ` ( P + 1 ) ) ) )
23	20, 22	sylibr 134	. . . . . . 7 |- ( ( ph /\ s e. NN ) -> DECID s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
24	23	ralrimiva 2615	. . . . . 6 |- ( ph -> A. s e. NN DECID s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
25		eleq1 2295	. . . . . . . 8 |- ( s = x -> ( s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) <-> x e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) ) )
26	25	dcbid 846	. . . . . . 7 |- ( s = x -> (DECID s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) <-> DECID x e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) ) )
27	26	cbvralvw 2782	. . . . . 6 |- ( A. s e. NN DECID s e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) <-> A. x e. NN DECID x e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
28	24, 27	sylib 122	. . . . 5 |- ( ph -> A. x e. NN DECID x e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
29		breq1 4112	. . . . . . . . 9 |- ( m = P -> ( m < n <-> P < n ) )
30	29	rexbidv 2543	. . . . . . . 8 |- ( m = P -> ( E. n e. A m < n <-> E. n e. A P < n ) )
31		nninfdclemf.nb	. . . . . . . 8 |- ( ph -> A. m e. NN E. n e. A m < n )
32	30, 31, 3	rspcdva 2926	. . . . . . 7 |- ( ph -> E. n e. A P < n )
33		breq2 4113	. . . . . . . 8 |- ( n = t -> ( P < n <-> P < t ) )
34	33	cbvrexvw 2783	. . . . . . 7 |- ( E. n e. A P < n <-> E. t e. A P < t )
35	32, 34	sylib 122	. . . . . 6 |- ( ph -> E. t e. A P < t )
36		simprl 531	. . . . . . . 8 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> t e. A )
37	3	nnzd 9699	. . . . . . . . . . 11 |- ( ph -> P e. ZZ )
38	37	peano2zd 9703	. . . . . . . . . 10 |- ( ph -> ( P + 1 ) e. ZZ )
39	38	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> ( P + 1 ) e. ZZ )
40	1	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> A C_ NN )
41	40, 36	sseldd 3239	. . . . . . . . . 10 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> t e. NN )
42	41	nnzd 9699	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> t e. ZZ )
43		simprr 533	. . . . . . . . . 10 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> P < t )
44		nnltp1le 9638	. . . . . . . . . . 11 |- ( ( P e. NN /\ t e. NN ) -> ( P < t <-> ( P + 1 ) <_ t ) )
45	3, 41, 44	syl2an2r 599	. . . . . . . . . 10 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> ( P < t <-> ( P + 1 ) <_ t ) )
46	43, 45	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> ( P + 1 ) <_ t )
47		eluz2 9859	. . . . . . . . 9 |- ( t e. ( ZZ>= ` ( P + 1 ) ) <-> ( ( P + 1 ) e. ZZ /\ t e. ZZ /\ ( P + 1 ) <_ t ) )
48	39, 42, 46, 47	syl3anbrc 1208	. . . . . . . 8 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> t e. ( ZZ>= ` ( P + 1 ) ) )
49	36, 48	elind 3404	. . . . . . 7 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> t e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
50		elex2 2830	. . . . . . 7 |- ( t e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) -> E. r r e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
51	49, 50	syl 14	. . . . . 6 |- ( ( ph /\ ( t e. A /\ P < t ) ) -> E. r r e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
52	35, 51	rexlimddv 2665	. . . . 5 |- ( ph -> E. r r e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
53		nnmindc 12730	. . . . 5 |- ( ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) C_ NN /\ A. x e. NN DECID x e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) /\ E. r r e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) ) -> inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
54	7, 28, 52, 53	syl3anc 1274	. . . 4 |- ( ph -> inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) e. ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
55	54	elin1d 3408	. . 3 |- ( ph -> inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) e. A )
56		fvoveq1 6073	. . . . . 6 |- ( y = P -> ( ZZ>= ` ( y + 1 ) ) = ( ZZ>= ` ( P + 1 ) ) )
57	56	ineq2d 3422	. . . . 5 |- ( y = P -> ( A i^i ( ZZ>= ` ( y + 1 ) ) ) = ( A i^i ( ZZ>= ` ( P + 1 ) ) ) )
58	57	infeq1d 7303	. . . 4 |- ( y = P -> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) = inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) )
59		eqidd 2233	. . . 4 |- ( z = Q -> inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) = inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) )
60		eqid 2232	. . . 4 |- ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) = ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) )
61	58, 59, 60	ovmpog 6188	. . 3 |- ( ( P e. NN /\ Q e. NN /\ inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) e. A ) -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) = inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) )
62	3, 5, 55, 61	syl3anc 1274	. 2 |- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) = inf ( ( A i^i ( ZZ>= ` ( P + 1 ) ) ) , RR , < ) )
63	62, 55	eqeltrd 2309	1 |- ( ph -> ( P ( y e. NN , z e. NN |-> inf ( ( A i^i ( ZZ>= ` ( y + 1 ) ) ) , RR , < ) ) Q ) e. A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   E.wrex 2521   i^i cin 3210   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   e. cmpo 6052  infcinf 7274   RRcr 8126   1c1 8128   + caddc 8130   < clt 8308   <_ cle 8309   NNcn 9237   ZZcz 9577   ZZ>=cuz 9853

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477

This theorem is referenced by:  nninfdclemf  13200  nninfdclemp1  13201