Theorem iseqf1olemklt 9910

Description: Lemma for seq3f1o 9929. (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 2276	. 2 |- ( ph -> -. K = ( `' J ` K ) )
3		iseqf1olemklt.j	. . . . . 6 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
4	3	adantr 270	. . . . 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 270	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ( M ... N ) )
7		f1ocnvfv2 5557	. . . . 5 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
8	4, 6, 7	syl2anc 403	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = K )
9		fveq2 5305	. . . . . 6 |- ( x = ( `' J ` K ) -> ( J ` x ) = ( J ` ( `' J ` K ) ) )
10		id 19	. . . . . 6 |- ( x = ( `' J ` K ) -> x = ( `' J ` K ) )
11	9, 10	eqeq12d 2102	. . . . 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 270	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> A. x e. ( M..^ K ) ( J ` x ) = x )
14		f1ocnv 5266	. . . . . . . . . . 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 5253	. . . . . . . . . 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	ffvelrnd 5435	. . . . . . . 8 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
19		elfzuz 9434	. . . . . . . 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 270	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( ZZ>= ` M ) )
22		elfzelz 9438	. . . . . . . 8 |- ( K e. ( M ... N ) -> K e. ZZ )
23	5, 22	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
24	23	adantr 270	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> K e. ZZ )
25		simpr 108	. . . . . 6 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) < K )
26		elfzo2 9557	. . . . . 6 |- ( ( `' J ` K ) e. ( M..^ K ) <-> ( ( `' J ` K ) e. ( ZZ>= ` M ) /\ K e. ZZ /\ ( `' J ` K ) < K ) )
27	21, 24, 25, 26	syl3anbrc 1127	. . . . 5 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( `' J ` K ) e. ( M..^ K ) )
28	11, 13, 27	rspcdva 2727	. . . 4 |- ( ( ph /\ ( `' J ` K ) < K ) -> ( J ` ( `' J ` K ) ) = ( `' J ` K ) )
29	8, 28	eqtr3d 2122	. . 3 |- ( ( ph /\ ( `' J ` K ) < K ) -> K = ( `' J ` K ) )
30	2, 29	mtand 626	. 2 |- ( ph -> -. ( `' J ` K ) < K )
31		elfzelz 9438	. . . 4 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
32	18, 31	syl 14	. . 3 |- ( ph -> ( `' J ` K ) e. ZZ )
33		ztri3or 8791	. . 3 |- ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ ) -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
34	23, 32, 33	syl2anc 403	. 2 |- ( ph -> ( K < ( `' J ` K ) \/ K = ( `' J ` K ) \/ ( `' J ` K ) < K ) )
35	2, 30, 34	ecase23d 1286	1 |- ( ph -> K < ( `' J ` K ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ w3o 923   = wceq 1289   e. wcel 1438   =/= wne 2255   A.wral 2359   class class class wbr 3845   `'ccnv 4437   -->wf 5011   -1-1-onto->wf1o 5014   ` cfv 5015  (class class class)co 5652   < clt 7520   ZZcz 8748   ZZ>=cuz 9017   ...cfz 9422  ..^cfzo 9549

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-ltadd 7459

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-fz 9423  df-fzo 9550

This theorem is referenced by:  seq3f1olemqsumkj  9923