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Theorem erdsze2lem1 31185
Description: Lemma for erdsze2 31187. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze2.r |- ( ph -> R e. NN )
erdsze2.s |- ( ph -> S e. NN )
erdsze2.f |- ( ph -> F : A -1-1-> RR )
erdsze2.a |- ( ph -> A C_ RR )
erdsze2lem.n |- N = ( ( R - 1 ) x. ( S - 1 ) )
erdsze2lem.l |- ( ph -> N < ( # ` A ) )
Assertion
Ref Expression
erdsze2lem1 |- ( ph -> E. f ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
Distinct variable groups:   A, f   f, F   R, f   S, f   f, N   ph, f
Proof of Theorem erdsze2lem1
Dummy variable s is distinct from all other variables.
StepHypRef Expression
1 erdsze2lem.n . . . . . . . . 9 |- N = ( ( R - 1 ) x. ( S - 1 ) )
2 erdsze2.r . . . . . . . . . . 11 |- ( ph -> R e. NN )
3 nnm1nn0 11334 . . . . . . . . . . 11 |- ( R e. NN -> ( R - 1 ) e. NN0 )
42, 3syl 17 . . . . . . . . . 10 |- ( ph -> ( R - 1 ) e. NN0 )
5 erdsze2.s . . . . . . . . . . 11 |- ( ph -> S e. NN )
6 nnm1nn0 11334 . . . . . . . . . . 11 |- ( S e. NN -> ( S - 1 ) e. NN0 )
75, 6syl 17 . . . . . . . . . 10 |- ( ph -> ( S - 1 ) e. NN0 )
84, 7nn0mulcld 11356 . . . . . . . . 9 |- ( ph -> ( ( R - 1 ) x. ( S - 1 ) ) e. NN0 )
91, 8syl5eqel 2705 . . . . . . . 8 |- ( ph -> N e. NN0 )
10 peano2nn0 11333 . . . . . . . 8 |- ( N e. NN0 -> ( N + 1 ) e. NN0 )
11 hashfz1 13134 . . . . . . . 8 |- ( ( N + 1 ) e. NN0 -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
129, 10, 113syl 18 . . . . . . 7 |- ( ph -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
1312adantr 481 . . . . . 6 |- ( ( ph /\ A e. Fin ) -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
14 erdsze2lem.l . . . . . . . 8 |- ( ph -> N < ( # ` A ) )
1514adantr 481 . . . . . . 7 |- ( ( ph /\ A e. Fin ) -> N < ( # ` A ) )
16 hashcl 13147 . . . . . . . 8 |- ( A e. Fin -> ( # ` A ) e. NN0 )
17 nn0ltp1le 11435 . . . . . . . 8 |- ( ( N e. NN0 /\ ( # ` A ) e. NN0 ) -> ( N < ( # ` A ) <-> ( N + 1 ) <_ ( # ` A ) ) )
189, 16, 17syl2an 494 . . . . . . 7 |- ( ( ph /\ A e. Fin ) -> ( N < ( # ` A ) <-> ( N + 1 ) <_ ( # ` A ) ) )
1915, 18mpbid 222 . . . . . 6 |- ( ( ph /\ A e. Fin ) -> ( N + 1 ) <_ ( # ` A ) )
2013, 19eqbrtrd 4675 . . . . 5 |- ( ( ph /\ A e. Fin ) -> ( # ` ( 1 ... ( N + 1 ) ) ) <_ ( # ` A ) )
21 fzfid 12772 . . . . . 6 |- ( ( ph /\ A e. Fin ) -> ( 1 ... ( N + 1 ) ) e. Fin )
22 simpr 477 . . . . . 6 |- ( ( ph /\ A e. Fin ) -> A e. Fin )
23 hashdom 13168 . . . . . 6 |- ( ( ( 1 ... ( N + 1 ) ) e. Fin /\ A e. Fin ) -> ( ( # ` ( 1 ... ( N + 1 ) ) ) <_ ( # ` A ) <-> ( 1 ... ( N + 1 ) ) ~<_ A ) )
2421, 22, 23syl2anc 693 . . . . 5 |- ( ( ph /\ A e. Fin ) -> ( ( # ` ( 1 ... ( N + 1 ) ) ) <_ ( # ` A ) <-> ( 1 ... ( N + 1 ) ) ~<_ A ) )
2520, 24mpbid 222 . . . 4 |- ( ( ph /\ A e. Fin ) -> ( 1 ... ( N + 1 ) ) ~<_ A )
26 simpr 477 . . . . . 6 |- ( ( ph /\ -. A e. Fin ) -> -. A e. Fin )
27 fzfid 12772 . . . . . 6 |- ( ( ph /\ -. A e. Fin ) -> ( 1 ... ( N + 1 ) ) e. Fin )
28 isinffi 8818 . . . . . 6 |- ( ( -. A e. Fin /\ ( 1 ... ( N + 1 ) ) e. Fin ) -> E. f f : ( 1 ... ( N + 1 ) ) -1-1-> A )
2926, 27, 28syl2anc 693 . . . . 5 |- ( ( ph /\ -. A e. Fin ) -> E. f f : ( 1 ... ( N + 1 ) ) -1-1-> A )
30 erdsze2.a . . . . . . . 8 |- ( ph -> A C_ RR )
31 reex 10027 . . . . . . . 8 |- RR e. _V
32 ssexg 4804 . . . . . . . 8 |- ( ( A C_ RR /\ RR e. _V ) -> A e. _V )
3330, 31, 32sylancl 694 . . . . . . 7 |- ( ph -> A e. _V )
3433adantr 481 . . . . . 6 |- ( ( ph /\ -. A e. Fin ) -> A e. _V )
35 brdomg 7965 . . . . . 6 |- ( A e. _V -> ( ( 1 ... ( N + 1 ) ) ~<_ A <-> E. f f : ( 1 ... ( N + 1 ) ) -1-1-> A ) )
3634, 35syl 17 . . . . 5 |- ( ( ph /\ -. A e. Fin ) -> ( ( 1 ... ( N + 1 ) ) ~<_ A <-> E. f f : ( 1 ... ( N + 1 ) ) -1-1-> A ) )
3729, 36mpbird 247 . . . 4 |- ( ( ph /\ -. A e. Fin ) -> ( 1 ... ( N + 1 ) ) ~<_ A )
3825, 37pm2.61dan 832 . . 3 |- ( ph -> ( 1 ... ( N + 1 ) ) ~<_ A )
39 domeng 7969 . . . 4 |- ( A e. _V -> ( ( 1 ... ( N + 1 ) ) ~<_ A <-> E. s ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) )
4033, 39syl 17 . . 3 |- ( ph -> ( ( 1 ... ( N + 1 ) ) ~<_ A <-> E. s ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) )
4138, 40mpbid 222 . 2 |- ( ph -> E. s ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) )
42 simprr 796 . . . . . 6 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> s C_ A )
4330adantr 481 . . . . . 6 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> A C_ RR )
4442, 43sstrd 3613 . . . . 5 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> s C_ RR )
45 ltso 10118 . . . . 5 |- < Or RR
46 soss 5053 . . . . 5 |- ( s C_ RR -> ( < Or RR -> < Or s ) )
4744, 45, 46mpisyl 21 . . . 4 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> < Or s )
48 fzfid 12772 . . . . 5 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( 1 ... ( N + 1 ) ) e. Fin )
49 simprl 794 . . . . . 6 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( 1 ... ( N + 1 ) ) ~~ s )
50 enfi 8176 . . . . . 6 |- ( ( 1 ... ( N + 1 ) ) ~~ s -> ( ( 1 ... ( N + 1 ) ) e. Fin <-> s e. Fin ) )
5149, 50syl 17 . . . . 5 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( ( 1 ... ( N + 1 ) ) e. Fin <-> s e. Fin ) )
5248, 51mpbid 222 . . . 4 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> s e. Fin )
53 fz1iso 13246 . . . 4 |- ( ( < Or s /\ s e. Fin ) -> E. f f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) )
5447, 52, 53syl2anc 693 . . 3 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> E. f f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) )
55 isof1o 6573 . . . . . . . . . 10 |- ( f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) -> f : ( 1 ... ( # ` s ) ) -1-1-onto-> s )
5655adantl 482 . . . . . . . . 9 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f : ( 1 ... ( # ` s ) ) -1-1-onto-> s )
57 hashen 13135 . . . . . . . . . . . . . . 15 |- ( ( ( 1 ... ( N + 1 ) ) e. Fin /\ s e. Fin ) -> ( ( # ` ( 1 ... ( N + 1 ) ) ) = ( # ` s ) <-> ( 1 ... ( N + 1 ) ) ~~ s ) )
5848, 52, 57syl2anc 693 . . . . . . . . . . . . . 14 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( ( # ` ( 1 ... ( N + 1 ) ) ) = ( # ` s ) <-> ( 1 ... ( N + 1 ) ) ~~ s ) )
5949, 58mpbird 247 . . . . . . . . . . . . 13 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( # ` s ) )
6012adantr 481 . . . . . . . . . . . . 13 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( # ` ( 1 ... ( N + 1 ) ) ) = ( N + 1 ) )
6159, 60eqtr3d 2658 . . . . . . . . . . . 12 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( # ` s ) = ( N + 1 ) )
6261adantr 481 . . . . . . . . . . 11 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( # ` s ) = ( N + 1 ) )
6362oveq2d 6666 . . . . . . . . . 10 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( 1 ... ( # ` s ) ) = ( 1 ... ( N + 1 ) ) )
64 f1oeq2 6128 . . . . . . . . . 10 |- ( ( 1 ... ( # ` s ) ) = ( 1 ... ( N + 1 ) ) -> ( f : ( 1 ... ( # ` s ) ) -1-1-onto-> s <-> f : ( 1 ... ( N + 1 ) ) -1-1-onto-> s ) )
6563, 64syl 17 . . . . . . . . 9 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( f : ( 1 ... ( # ` s ) ) -1-1-onto-> s <-> f : ( 1 ... ( N + 1 ) ) -1-1-onto-> s ) )
6656, 65mpbid 222 . . . . . . . 8 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f : ( 1 ... ( N + 1 ) ) -1-1-onto-> s )
67 f1of1 6136 . . . . . . . 8 |- ( f : ( 1 ... ( N + 1 ) ) -1-1-onto-> s -> f : ( 1 ... ( N + 1 ) ) -1-1-> s )
6866, 67syl 17 . . . . . . 7 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f : ( 1 ... ( N + 1 ) ) -1-1-> s )
69 simplrr 801 . . . . . . 7 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> s C_ A )
70 f1ss 6106 . . . . . . 7 |- ( ( f : ( 1 ... ( N + 1 ) ) -1-1-> s /\ s C_ A ) -> f : ( 1 ... ( N + 1 ) ) -1-1-> A )
7168, 69, 70syl2anc 693 . . . . . 6 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f : ( 1 ... ( N + 1 ) ) -1-1-> A )
72 simpr 477 . . . . . . . 8 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) )
73 f1ofo 6144 . . . . . . . . 9 |- ( f : ( 1 ... ( # ` s ) ) -1-1-onto-> s -> f : ( 1 ... ( # ` s ) ) -onto-> s )
74 forn 6118 . . . . . . . . 9 |- ( f : ( 1 ... ( # ` s ) ) -onto-> s -> ran f = s )
75 isoeq5 6571 . . . . . . . . 9 |- ( ran f = s -> ( f Isom < , < ( ( 1 ... ( # ` s ) ) , ran f ) <-> f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) )
7656, 73, 74, 754syl 19 . . . . . . . 8 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( f Isom < , < ( ( 1 ... ( # ` s ) ) , ran f ) <-> f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) )
7772, 76mpbird 247 . . . . . . 7 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f Isom < , < ( ( 1 ... ( # ` s ) ) , ran f ) )
78 isoeq4 6570 . . . . . . . 8 |- ( ( 1 ... ( # ` s ) ) = ( 1 ... ( N + 1 ) ) -> ( f Isom < , < ( ( 1 ... ( # ` s ) ) , ran f ) <-> f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
7963, 78syl 17 . . . . . . 7 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( f Isom < , < ( ( 1 ... ( # ` s ) ) , ran f ) <-> f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
8077, 79mpbid 222 . . . . . 6 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) )
8171, 80jca 554 . . . . 5 |- ( ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) /\ f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) ) -> ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
8281ex 450 . . . 4 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) -> ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) ) )
8382eximdv 1846 . . 3 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> ( E. f f Isom < , < ( ( 1 ... ( # ` s ) ) , s ) -> E. f ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) ) )
8454, 83mpd 15 . 2 |- ( ( ph /\ ( ( 1 ... ( N + 1 ) ) ~~ s /\ s C_ A ) ) -> E. f ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
8541, 84exlimddv 1863 1 |- ( ph -> E. f ( f : ( 1 ... ( N + 1 ) ) -1-1-> A /\ f Isom < , < ( ( 1 ... ( N + 1 ) ) , ran f ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Or wor 5034   ran crn 5115   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ...cfz 12326   #chash 13117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118
This theorem is referenced by:  erdsze2  31187
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