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Theorem zfz1isolemiso 10948
Description: Lemma for zfz1iso 10950. 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 5858 . . . . 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 1247 . . . 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 10117 . . . . . . . . . . 11 |- ( A e. ( 1 ... (♯ ` X ) ) -> A e. ZZ )
107, 9syl 14 . . . . . . . . . 10 |- ( ph -> A e. ZZ )
1110zred 9465 . . . . . . . . 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 10890 . . . . . . . . . . . . 13 |- ( X e. Fin -> (♯ ` X ) e. NN0 )
1513, 14syl 14 . . . . . . . . . . . 12 |- ( ph -> (♯ ` X ) e. NN0 )
1615nn0red 9320 . . . . . . . . . . 11 |- ( ph -> (♯ ` X ) e. RR )
17 peano2rem 8310 . . . . . . . . . . 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 10120 . . . . . . . . . . 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 10928 . . . . . . . . . . . 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 4060 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A <_ ( (♯ ` X ) - 1 ) )
2820ltm1d 8976 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( (♯ ` X ) - 1 ) < (♯ ` X ) )
2912, 19, 20, 27, 28lelttrd 8168 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A < (♯ ` X ) )
3012, 29gtned 8156 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) =/= A )
31 fvunsng 5759 . . . . . . 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 10117 . . . . . . . . . 10 |- ( B e. ( 1 ... (♯ ` X ) ) -> B e. ZZ )
3734, 36syl 14 . . . . . . . . 9 |- ( ph -> B e. ZZ )
3837zred 9465 . . . . . . . 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 10120 . . . . . . . . . 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 4060 . . . . . . . 8 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ ( (♯ ` X ) - 1 ) )
4616ltm1d 8976 . . . . . . . . 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 8168 . . . . . . 7 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B < (♯ ` X ) )
4939, 48gtned 8156 . . . . . 6 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> (♯ ` X ) =/= B )
50 fvunsng 5759 . . . . . 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 4047 . . . 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 3641 . . . . . 6 |- ( B e. { (♯ ` X ) } -> B = (♯ ` X ) )
5655adantl 277 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> B = (♯ ` X ) )
5754, 56breqtrrd 4062 . . . 4 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> A < B )
58 isof1o 5857 . . . . . . . . . . . . 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 5507 . . . . . . . . . . . 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 5700 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. ( X \ { M } ) )
6362eldifbd 3169 . . . . . . . . 9 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( G ` A ) e. { M } )
64 elsn2g 3656 . . . . . . . . . . 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 673 . . . . . . . 8 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( G ` A ) = M )
68 breq1 4037 . . . . . . . . . 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 3168 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. X )
7268, 70, 71rspcdva 2873 . . . . . . . . 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 3185 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) e. ZZ )
7673, 23sseldd 3185 . . . . . . . . . . 11 |- ( ph -> M e. ZZ )
7776adantr 276 . . . . . . . . . 10 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> M e. ZZ )
78 zleloe 9390 . . . . . . . . . 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 1360 . . . . . . 7 |- ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` A ) < M )
8215nn0zd 9463 . . . . . . . . . . . . . . 15 |- ( ph -> (♯ ` X ) e. ZZ )
83 peano2zm 9381 . . . . . . . . . . . . . . 15 |- ( (♯ ` X ) e. ZZ -> ( (♯ ` X ) - 1 ) e. ZZ )
8482, 83syl 14 . . . . . . . . . . . . . 14 |- ( ph -> ( (♯ ` X ) - 1 ) e. ZZ )
85 zltnle 9389 . . . . . . . . . . . . . 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 4046 . . . . . . . . . . . 12 |- ( ph -> ( (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) <-> (♯ ` X ) <_ ( (♯ ` X ) - 1 ) ) )
8987, 88mtbird 674 . . . . . . . . . . 11 |- ( ph -> -. (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) )
90 elfzle2 10120 . . . . . . . . . . 11 |- ( (♯ ` X ) e. ( 1 ... (♯ ` ( X \ { M } ) ) ) -> (♯ ` X ) <_ (♯ ` ( X \ { M } ) ) )
9189, 90nsyl 629 . . . . . . . . . 10 |- ( ph -> -. (♯ ` X ) e. ( 1 ... (♯ ` ( X \ { M } ) ) ) )
92 fdm 5416 . . . . . . . . . . 11 |- ( G : ( 1 ... (♯ ` ( X \ { M } ) ) ) --> ( X \ { M } ) -> dom G = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
9361, 92syl 14 . . . . . . . . . 10 |- ( ph -> dom G = ( 1 ... (♯ ` ( X \ { M } ) ) ) )
9491, 93neleqtrrd 2295 . . . . . . . . 9 |- ( ph -> -. (♯ ` X ) e. dom G )
95 fsnunfv 5766 . . . . . . . . 9 |- ( ( (♯ ` X ) e. NN0 /\ M e. X /\ -. (♯ ` X ) e. dom G ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) = M )
9615, 23, 94, 95syl3anc 1249 . . . . . . . 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 4066 . . . . . 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 5565 . . . . 5 |- ( ( ( ph /\ A e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
10199, 100breqtrrd 4062 . . . 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 10947 . . . . . 6 |- ( ph -> ( 1 ... (♯ ` X ) ) = ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
10434, 103eleqtrd 2275 . . . . 5 |- ( ph -> B e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
105 elun 3305 . . . . 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 799 . 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 10120 . . . . . . . 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 3641 . . . . . . 7 |- ( A e. { (♯ ` X ) } -> A = (♯ ` X ) )
115114ad2antlr 489 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> A = (♯ ` X ) )
116113, 115breqtrrd 4062 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B <_ A )
117109, 110, 116lensymd 8165 . . . 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 5701 . . . . . . . . 9 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. ( X \ { M } ) )
122121eldifad 3168 . . . . . . . 8 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. X )
123118, 122sseldd 3185 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. ZZ )
124123zred 9465 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) e. RR )
12576zred 9465 . . . . . . 7 |- ( ph -> M e. RR )
126125ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> M e. RR )
127 breq1 4037 . . . . . . 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 2873 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> ( G ` B ) <_ M )
130124, 126, 129lensymd 8165 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. M < ( G ` B ) )
131115fveq2d 5565 . . . . . . 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 2229 . . . . . 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 4060 . . . . . . . . 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 8168 . . . . . . . 8 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> B < (♯ ` X ) )
142109, 141gtned 8156 . . . . . . 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 4047 . . . . 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 674 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. ( 1 ... (♯ ` ( X \ { M } ) ) ) ) -> -. ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) )
146117, 1452falsed 703 . . 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 8155 . . . . 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 2232 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> A = B )
152151breq1d 4044 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( A < B <-> B < B ) )
153148, 152mtbird 674 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. A < B )
154125ad2antrr 488 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> M e. RR )
155154ltnrd 8155 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. M < M )
156149fveq2d 5565 . . . . . . 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 2229 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) = M )
159150fveq2d 5565 . . . . . . 7 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = ( ( G u. { <. (♯ ` X ) , M >. } ) ` (♯ ` X ) ) )
160159, 157eqtrd 2229 . . . . . 6 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) = M )
161158, 160breq12d 4047 . . . . 5 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> ( ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) <-> M < M ) )
162155, 161mtbird 674 . . . 4 |- ( ( ( ph /\ A e. { (♯ ` X ) } ) /\ B e. { (♯ ` X ) } ) -> -. ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) )
163153, 1622falsed 703 . . 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 799 . 2 |- ( ( ph /\ A e. { (♯ ` X ) } ) -> ( A < B <-> ( ( G u. { <. (♯ ` X ) , M >. } ) ` A ) < ( ( G u. { <. (♯ ` X ) , M >. } ) ` B ) ) )
1667, 103eleqtrd 2275 . . 3 |- ( ph -> A e. ( ( 1 ... (♯ ` ( X \ { M } ) ) ) u. { (♯ ` X ) } ) )
167 elun 3305 . . 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 799 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 709   = wceq 1364   e. wcel 2167   =/= wne 2367   A.wral 2475   \ cdif 3154   u. cun 3155   C_ wss 3157   {csn 3623   <.cop 3626   class class class wbr 4034   dom cdm 4664   -->wf 5255   -1-1-onto->wf1o 5258   ` cfv 5259   Isom wiso 5260  (class class class)co 5925   Fincfn 6808   RRcr 7895   1c1 7897   < clt 8078   <_ cle 8079   - cmin 8214   NN0cn0 9266   ZZcz 9343   ...cfz 10100  ♯chash 10884
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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-ihash 10885
This theorem is referenced by:  zfz1isolem1  10949
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