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Theorem zfz1isolemiso 11061
Description: Lemma for zfz1iso 11063. Adding one element to the order isomorphism. (Contributed by Jim Kingdon, 8-Sep-2022.)
Hypotheses
Ref Expression
zfz1isolemiso.xf |- ( ph -> X e. Fin )
zfz1isolemiso.xz |- ( ph -> X C_ ZZ )
zfz1isolemiso.mx |- ( ph -> M e. X )
zfz1isolemiso.m |- ( ph -> A. z e. X z <_ M )
zfz1isolemiso.g |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
zfz1isolemiso.a |- ( ph -> A e. ( 1 ... (♯ ` X ) ) )
zfz1isolemiso.b |- ( ph -> B e. ( 1 ... (♯ ` X ) ) )
Assertion
Ref Expression
zfz1isolemiso |- ( ph -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
Distinct variable groups:   z, A   z, B   z, G   z, M   z, X
Allowed substitution hint:   ph( z)
Proof of Theorem zfz1isolemiso
StepHypRef Expression
1 zfz1isolemiso.g . . . . . 6 |- ( ph -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
21ad2antrr 488 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) )
3 simplr 528 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) )
4 simpr 110 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) )
5 isorel 5932 . . . . 5 |- ( ( G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) /\ ( A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) ) -> ( A < B <-> ( G ` A ) < ( G ` B ) ) )
62, 3, 4, 5syl12anc 1269 . . . 4 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( A < B <-> ( G ` A ) < ( G ` B ) ) )
7 zfz1isolemiso.a . . . . . . . 8 |- ( ph -> A e. ( 1 ... (♯ ` X ) ) )
87adantr 276 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A e. ( 1 ... (♯ ` X ) ) )
9 elfzelz 10221 . . . . . . . . . . 11 |- ( A e. ( 1 ... (♯ ` X ) ) -> A e. ZZ )
107, 9syl 14 . . . . . . . . . 10 |- ( ph -> A e. ZZ )
1110zred 9569 . . . . . . . . 9 |- ( ph -> A e. RR )
1211adantr 276 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A e. RR )
13 zfz1isolemiso.xf . . . . . . . . . . . . 13 |- ( ph -> X e. Fin )
14 hashcl 11003 . . . . . . . . . . . . 13 |- ( X e. Fin -> (♯ ` X ) e. NN0 )
1513, 14syl 14 . . . . . . . . . . . 12 |- ( ph -> (♯ ` X ) e. NN0 )
1615nn0red 9423 . . . . . . . . . . 11 |- ( ph -> (♯ ` X ) e. RR )
17 peano2rem 8413 . . . . . . . . . . 11 |- ( (♯ ` X ) e. RR -> ( (♯ ` X ) - 1 ) e. RR )
1816, 17syl 14 . . . . . . . . . 10 |- ( ph -> ( (♯ ` X ) - 1 ) e. RR )
1918adantr 276 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) e. RR )
2016adantr 276 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) e. RR )
21 elfzle2 10224 . . . . . . . . . . 11 |- ( A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) -> A <_ (♯ ` ( X \ { M } ) ) )
2221adantl 277 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A <_ (♯ ` ( X \ { M } ) ) )
23 zfz1isolemiso.mx . . . . . . . . . . . 12 |- ( ph -> M e. X )
24 hashdifsn 11041 . . . . . . . . . . . 12 |- ( ( X e. Fin /\ M e. X ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
2513, 23, 24syl2anc 411 . . . . . . . . . . 11 |- ( ph -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
2625adantr 276 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
2722, 26breqtrd 4109 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A <_ ( (♯ ` X ) - 1 ) )
2820ltm1d 9079 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) < (♯ ` X ) )
2912, 19, 20, 27, 28lelttrd 8271 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A < (♯ ` X ) )
3012, 29gtned 8259 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) =/= A )
31 fvunsng 5833 . . . . . . 7 |- ( ( A e. ( 1 ... (♯ ` X ) ) /\ (♯ ` X ) =/= A ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = ( G ` A ) )
328, 30, 31syl2anc 411 . . . . . 6 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = ( G ` A ) )
3332adantr 276 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = ( G ` A ) )
34 zfz1isolemiso.b . . . . . . 7 |- ( ph -> B e. ( 1 ... (♯ ` X ) ) )
3534ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. ( 1 ... (♯ ` X ) ) )
36 elfzelz 10221 . . . . . . . . . 10 |- ( B e. ( 1 ... (♯ ` X ) ) -> B e. ZZ )
3734, 36syl 14 . . . . . . . . 9 |- ( ph -> B e. ZZ )
3837zred 9569 . . . . . . . 8 |- ( ph -> B e. RR )
3938ad2antrr 488 . . . . . . 7 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. RR )
4018ad2antrr 488 . . . . . . . 8 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) e. RR )
4116ad2antrr 488 . . . . . . . 8 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) e. RR )
42 elfzle2 10224 . . . . . . . . . 10 |- ( B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) -> B <_ (♯ ` ( X \ { M } ) ) )
4342adantl 277 . . . . . . . . 9 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ (♯ ` ( X \ { M } ) ) )
4425ad2antrr 488 . . . . . . . . 9 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
4543, 44breqtrd 4109 . . . . . . . 8 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ ( (♯ ` X ) - 1 ) )
4616ltm1d 9079 . . . . . . . . 9 |- ( ph -> ( (♯ ` X ) - 1 ) < (♯ ` X ) )
4746ad2antrr 488 . . . . . . . 8 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) < (♯ ` X ) )
4839, 40, 41, 45, 47lelttrd 8271 . . . . . . 7 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B < (♯ ` X ) )
4939, 48gtned 8259 . . . . . 6 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) =/= B )
50 fvunsng 5833 . . . . . 6 |- ( ( B e. ( 1 ... (♯ ` X ) ) /\ (♯ ` X ) =/= B ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( G ` B ) )
5135, 49, 50syl2anc 411 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( G ` B ) )
5233, 51breq12d 4096 . . . 4 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) <-> ( G ` A ) < ( G ` B ) ) )
536, 52bitr4d 191 . . 3 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
5429adantr 276 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> A < (♯ ` X ) )
55 elsni 3684 . . . . . 6 |- ( B e. { (♯ ` X ) } -> B = (♯ ` X ) )
5655adantl 277 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> B = (♯ ` X ) )
5754, 56breqtrrd 4111 . . . 4 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> A < B )
58 isof1o 5931 . . . . . . . . . . . . 13 |- ( G Isom < , < ( ( 1 ... (♯ ` ( X \ { M } ) ) ) , ( X \ { M } ) ) -> G : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) )
591, 58syl 14 . . . . . . . . . . . 12 |- ( ph -> G : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) )
60 f1of 5572 . . . . . . . . . . . 12 |- ( G : ( 1 ... (♯ ` ( X \ { M } ) ) ) -1-1-onto-> ( X \ { M } ) -> G : ( 1 ... (♯ ` ( X \ { M } ) ) ) --> ( X \ { M } ) )
6159, 60syl 14 . . . . . . . . . . 11 |- ( ph -> G : ( 1 ... (♯ ` ( X \ { M } ) ) ) --> ( X \ { M } ) )
6261ffvelcdmda 5770 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. ( X \ { M } ) )
6362eldifbd 3209 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( G ` A ) e. { M } )
64 elsn2g 3699 . . . . . . . . . . 11 |- ( M e. X -> ( ( G ` A ) e. { M } <-> ( G ` A ) = M ) )
6523, 64syl 14 . . . . . . . . . 10 |- ( ph -> ( ( G ` A ) e. { M } <-> ( G ` A ) = M ) )
6665adantr 276 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G ` A ) e. { M } <-> ( G ` A ) = M ) )
6763, 66mtbid 676 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( G ` A ) = M )
68 breq1 4086 . . . . . . . . . 10 |- ( z = ( G ` A ) -> ( z <_ M <-> ( G ` A ) <_ M ) )
69 zfz1isolemiso.m . . . . . . . . . . 11 |- ( ph -> A. z e. X z <_ M )
7069adantr 276 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A. z e. X z <_ M )
7162eldifad 3208 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. X )
7268, 70, 71rspcdva 2912 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) <_ M )
73 zfz1isolemiso.xz . . . . . . . . . . . 12 |- ( ph -> X C_ ZZ )
7473adantr 276 . . . . . . . . . . 11 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> X C_ ZZ )
7574, 71sseldd 3225 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. ZZ )
7673, 23sseldd 3225 . . . . . . . . . . 11 |- ( ph -> M e. ZZ )
7776adantr 276 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> M e. ZZ )
78 zleloe 9493 . . . . . . . . . 10 |- ( ( ( G ` A ) e. ZZ /\ M e. ZZ ) -> ( ( G ` A ) <_ M <-> ( ( G ` A ) < M \/ ( G ` A ) = M ) ) )
7975, 77, 78syl2anc 411 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G ` A ) <_ M <-> ( ( G ` A ) < M \/ ( G ` A ) = M ) ) )
8072, 79mpbid 147 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G ` A ) < M \/ ( G ` A ) = M ) )
8167, 80ecased 1383 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) < M )
8215nn0zd 9567 . . . . . . . . . . . . . . 15 |- ( ph -> (♯ ` X ) e. ZZ )
83 peano2zm 9484 . . . . . . . . . . . . . . 15 |- ( (♯ ` X ) e. ZZ -> ( (♯ ` X ) - 1 ) e. ZZ )
8482, 83syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( (♯ ` X ) - 1 ) e. ZZ )
85 zltnle 9492 . . . . . . . . . . . . . 14 |- ( ( ( (♯ ` X ) - 1 ) e. ZZ /\ (♯ ` X ) e. ZZ ) -> ( ( (♯ ` X ) - 1 ) < (♯ ` X ) <-> -. (♯ ` X ) <_ ( (♯ ` X ) - 1 ) ) )
8684, 82, 85syl2anc 411 . . . . . . . . . . . . 13 |- ( ph -> ( ( (♯ ` X ) - 1 ) < (♯ ` X ) <-> -. (♯ ` X ) <_ ( (♯ ` X ) - 1 ) ) )
8746, 86mpbid 147 . . . . . . . . . . . 12 |- ( ph -> -. (♯ ` X ) <_ ( (♯ ` X ) - 1 ) )
8825breq2d 4095 . . . . . . . . . . . 12 |- ( ph -> ( (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) <-> (♯ ` X ) <_ ( (♯ ` X ) - 1 ) ) )
8987, 88mtbird 677 . . . . . . . . . . 11 |- ( ph -> -. (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) )
90 elfzle2 10224 . . . . . . . . . . 11 |- ( (♯ ` X ) e. ( 1 ... (♯ ` ( X \ { M } ) ) ) -> (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) )
9189, 90nsyl 631 . . . . . . . . . 10 |- ( ph -> -. (♯ ` X ) e. ( 1 ... (♯ ` ( X \ { M } ) ) ) )
92 fdm 5479 . . . . . . . . . . 11 |- ( G : ( 1 ... (♯ ` ( X \ { M } ) ) ) --> ( X \ { M } ) -> dom G = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
9361, 92syl 14 . . . . . . . . . 10 |- ( ph -> dom G = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
9491, 93neleqtrrd 2328 . . . . . . . . 9 |- ( ph -> -. (♯ ` X ) e. dom G )
95 fsnunfv 5840 . . . . . . . . 9 |- ( ( (♯ ` X ) e. NN0 /\ M e. X /\ -. (♯ ` X ) e. dom G ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
9615, 23, 94, 95syl3anc 1271 . . . . . . . 8 |- ( ph -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
9796adantr 276 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
9881, 32, 973brtr4d 4115 . . . . . 6 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
9998adantr 276 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
10056fveq2d 5631 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
10199, 100breqtrrd 4111 . . . 4 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) )
10257, 1012thd 175 . . 3 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
10313, 23zfz1isolemsplit 11060 . . . . . 6 |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
10434, 103eleqtrd 2308 . . . . 5 |- ( ph -> B e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
105 elun 3345 . . . . 5 |- ( B e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) <-> ( B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ B e. { (♯ ` X ) } ) )
106104, 105sylib 122 . . . 4 |- ( ph -> ( B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ B e. { (♯ ` X ) } ) )
107106adantr 276 . . 3 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ B e. { (♯ ` X ) } ) )
10853, 102, 107mpjaodan 803 . 2 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
10938ad2antrr 488 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. RR )
11011ad2antrr 488 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A e. RR )
111 elfzle2 10224 . . . . . . . 8 |- ( B e. ( 1 ... (♯ ` X ) ) -> B <_ (♯ ` X ) )
11234, 111syl 14 . . . . . . 7 |- ( ph -> B <_ (♯ ` X ) )
113112ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ (♯ ` X ) )
114 elsni 3684 . . . . . . 7 |- ( A e. { (♯ ` X ) } -> A = (♯ ` X ) )
115114ad2antlr 489 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A = (♯ ` X ) )
116113, 115breqtrrd 4111 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ A )
117109, 110, 116lensymd 8268 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. A < B )
11873ad2antrr 488 . . . . . . . 8 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> X C_ ZZ )
11961ad2antrr 488 . . . . . . . . . 10 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> G : ( 1 ... (♯ ` ( X \ { M } ) ) ) --> ( X \ { M } ) )
120 simpr 110 . . . . . . . . . 10 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) )
121119, 120ffvelcdmd 5771 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. ( X \ { M } ) )
122121eldifad 3208 . . . . . . . 8 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. X )
123118, 122sseldd 3225 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. ZZ )
124123zred 9569 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. RR )
12576zred 9569 . . . . . . 7 |- ( ph -> M e. RR )
126125ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> M e. RR )
127 breq1 4086 . . . . . . 7 |- ( z = ( G ` B ) -> ( z <_ M <-> ( G ` B ) <_ M ) )
12869ad2antrr 488 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A. z e. X z <_ M )
129127, 128, 122rspcdva 2912 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) <_ M )
130124, 126, 129lensymd 8268 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. M < ( G ` B ) )
131115fveq2d 5631 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
13296ad2antrr 488 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
133131, 132eqtrd 2262 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = M )
13434ad2antrr 488 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B e. ( 1 ... (♯ ` X ) ) )
13518ad2antrr 488 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) e. RR )
13616ad2antrr 488 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) e. RR )
13742adantl 277 . . . . . . . . . 10 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ (♯ ` ( X \ { M } ) ) )
13825ad2antrr 488 . . . . . . . . . 10 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` ( X \ { M } ) ) = ( (♯ ` X ) - 1 ) )
139137, 138breqtrd 4109 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ ( (♯ ` X ) - 1 ) )
14046ad2antrr 488 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) < (♯ ` X ) )
141109, 135, 136, 139, 140lelttrd 8271 . . . . . . . 8 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B < (♯ ` X ) )
142109, 141gtned 8259 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) =/= B )
143134, 142, 50syl2anc 411 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( G ` B ) )
144133, 143breq12d 4096 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) <-> M < ( G ` B ) ) )
145130, 144mtbird 677 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) )
146117, 1452falsed 707 . . 3 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
14738ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> B e. RR )
148147ltnrd 8258 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. B < B )
149114ad2antlr 489 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> A = (♯ ` X ) )
15055adantl 277 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> B = (♯ ` X ) )
151149, 150eqtr4d 2265 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> A = B )
152151breq1d 4093 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( A < B <-> B < B ) )
153148, 152mtbird 677 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. A < B )
154125ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> M e. RR )
155154ltnrd 8258 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. M < M )
156149fveq2d 5631 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
15796ad2antrr 488 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
158156, 157eqtrd 2262 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = M )
159150fveq2d 5631 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
160159, 157eqtrd 2262 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = M )
161158, 160breq12d 4096 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) <-> M < M ) )
162155, 161mtbird 677 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) )
163153, 1622falsed 707 . . 3 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
164106adantr 276 . . 3 |- ( ( ph /\ A e. { (♯ ` X ) } ) -> ( B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ B e. { (♯ ` X ) } ) )
165146, 163, 164mpjaodan 803 . 2 |- ( ( ph /\ A e. { (♯ ` X ) } ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
1667, 103eleqtrd 2308 . . 3 |- ( ph -> A e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
167 elun 3345 . . 3 |- ( A e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) <-> ( A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ A e. { (♯ ` X ) } ) )
168166, 167sylib 122 . 2 |- ( ph -> ( A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) \/ A e. { (♯ ` X ) } ) )
169108, 165, 168mpjaodan 803 1 |- ( ph -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   \ cdif 3194   u. cun 3195   C_ wss 3197   {csn 3666   <.cop 3669   class class class wbr 4083   dom cdm 4719   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318   Isom wiso 5319  (class class class)co 6001   Fincfn 6887   RRcr 7998   1c1 8000   < clt 8181   <_ cle 8182   - cmin 8317   NN0cn0 9369   ZZcz 9446   ...cfz 10204  ♯chash 10997
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447  df-uz 9723  df-fz 10205  df-ihash 10998
This theorem is referenced by:  zfz1isolem1  11062
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