Theorem iseqf1olemklt 10864

Description: Lemma for seq3f1o 10883. (Contributed by Jim Kingdon, 21-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemklt.n	|- ( ph -> N e. ( ZZ>= ` M ) )
iseqf1olemklt.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemklt.j	|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemklt.const	|- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )
iseqf1olemklt.kj	|- ( ph -> K =/= ( `' J ` K ) )

Assertion

Ref	Expression
iseqf1olemklt	|- ( ph -> K < ( `' J ` K ) )

Distinct variable groups:   x, J   x, K   x, M

Allowed substitution hints:   ph( x)   N( x)

Proof of Theorem iseqf1olemklt

Step	Hyp	Ref	Expression
1		iseqf1olemklt.kj	. . 3 |- ( ph -> K =/= ( `' J ` K ) )
2	1	neneqd 2435	. 2 |- ( ph -> -. K = ( `' J ` K ) )
3		iseqf1olemklt.j	. . . . . 6 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4	3	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
5		iseqf1olemklt.k	. . . . . 6 |- ( ph -> K e. ( M ... N ) )
6	5	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ( M ... N ) )
7		f1ocnvfv2 5953	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
8	4, 6, 7	syl2anc 411	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
9		fveq2 5672	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
10		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
11	9, 10	eqeq12d 2249	. . . . 5 |- ( x = ( `' J ` K ) -> ( ( J ` x ) = x <-> ( J ` ( `' J ` K ) ) = ( `' J ` K ) ) )
12		iseqf1olemklt.const	. . . . . 6 |- ( ph -> A. x e. ( M..^ K ) ( J ` x ) = x )
13	12	adantr 276	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> A. x e. ( M..^ K ) ( J ` x ) = x )
14		f1ocnv 5629	. . . . . . . . . . 11 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
15	3, 14	syl 14	. . . . . . . . . 10 |- ( ph -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
16		f1of 5616	. . . . . . . . . 10 |- ( `' J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) --> ( M ... N ) )
17	15, 16	syl 14	. . . . . . . . 9 |- ( ph -> `' J : ( M ... N ) --> ( M ... N ) )
18	17, 5	ffvelcdmd 5815	. . . . . . . 8 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
19		elfzuz 10358	. . . . . . . 8 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
20	18, 19	syl 14	. . . . . . 7 |- ( ph -> ( `' J ` K ) e. ( ZZ>= ` M ) )
21	20	adantr 276	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
22		elfzelz 10362	. . . . . . . 8 |- ( K e. ( M ... N ) -> K e. ZZ )
23	5, 22	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
24	23	adantr 276	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ZZ )
25		simpr 110	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) < K )
26		elfzo2 10488	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
27	21, 24, 25, 26	syl3anbrc 1208	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
28	11, 13, 27	rspcdva 2928	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
29	8, 28	eqtr3d 2269	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
30	2, 29	mtand 671	. 2 |- ( ph -> -. ( `' J ` K ) < K )
31		elfzelz 10362	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
32	18, 31	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
33		ztri3or 9622	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
34	23, 32, 33	syl2anc 411	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
35	2, 30, 34	ecase23d 1387	1 |- ( ph -> K < ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ w3o 1004   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   class class class wbr 4111   `'ccnv 4750   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052   < clt 8310   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345  ..^cfzo 10480

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-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  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-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-fzo 10481

This theorem is referenced by:  seq3f1olemqsumkj  10877