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Theorem fsuppmapnn0fiublem 12076
Description: Lemma for fsuppmapnn0fiub 12077 and fsuppmapnn0fiubex 12078. (Contributed by AV, 2-Oct-2019.)
Hypotheses
Ref Expression
fsuppmapnn0fiub.u  |-  U  = 
U_ f  e.  M  ( f supp  Z )
fsuppmapnn0fiub.s  |-  S  =  sup ( U ,  RR ,  <  )
Assertion
Ref Expression
fsuppmapnn0fiublem  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )  ->  S  e.  NN0 ) )
Distinct variable groups:    f, M    R, f    U, f    f, V   
f, Z
Allowed substitution hint:    S( f)

Proof of Theorem fsuppmapnn0fiublem
StepHypRef Expression
1 fsuppmapnn0fiub.u . . . 4  |-  U  = 
U_ f  e.  M  ( f supp  Z )
2 nfv 1683 . . . . . . 7  |-  F/ f ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )
3 nfra1 2848 . . . . . . . 8  |-  F/ f A. f  e.  M  f finSupp  Z
4 nfv 1683 . . . . . . . 8  |-  F/ f  U  =/=  (/)
53, 4nfan 1875 . . . . . . 7  |-  F/ f ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )
62, 5nfan 1875 . . . . . 6  |-  F/ f ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )
7 suppssdm 6926 . . . . . . . 8  |-  ( f supp 
Z )  C_  dom  f
8 ssel2 3504 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
9 elmapfn 7453 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
10 fndm 5686 . . . . . . . . . . . . . 14  |-  ( f  Fn  NN0  ->  dom  f  =  NN0 )
11 eqimss 3561 . . . . . . . . . . . . . 14  |-  ( dom  f  =  NN0  ->  dom  f  C_  NN0 )
1210, 11syl 16 . . . . . . . . . . . . 13  |-  ( f  Fn  NN0  ->  dom  f  C_ 
NN0 )
138, 9, 123syl 20 . . . . . . . . . . . 12  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  dom  f  C_  NN0 )
1413ex 434 . . . . . . . . . . 11  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  dom  f  C_ 
NN0 ) )
15143ad2ant1 1017 . . . . . . . . . 10  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  ->  dom  f  C_  NN0 )
)
1615adantr 465 . . . . . . . . 9  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  (
f  e.  M  ->  dom  f  C_  NN0 )
)
1716imp 429 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  /\  f  e.  M )  ->  dom  f  C_  NN0 )
187, 17syl5ss 3520 . . . . . . 7  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  /\  f  e.  M )  ->  ( f supp  Z ) 
C_  NN0 )
1918ex 434 . . . . . 6  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  (
f  e.  M  -> 
( f supp  Z ) 
C_  NN0 ) )
206, 19ralrimi 2867 . . . . 5  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  A. f  e.  M  ( f supp  Z )  C_  NN0 )
21 iunss 4372 . . . . 5  |-  ( U_ f  e.  M  (
f supp  Z )  C_  NN0  <->  A. f  e.  M  ( f supp  Z )  C_  NN0 )
2220, 21sylibr 212 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U_ f  e.  M  ( f supp  Z )  C_  NN0 )
231, 22syl5eqss 3553 . . 3  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U  C_ 
NN0 )
24 ltso 9677 . . . . 5  |-  <  Or  RR
2524a1i 11 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  <  Or  RR )
26 simp2 997 . . . . . 6  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  M  e.  Fin )
27 id 22 . . . . . . . . 9  |-  ( f finSupp  Z  ->  f finSupp  Z )
2827fsuppimpd 7848 . . . . . . . 8  |-  ( f finSupp  Z  ->  ( f supp  Z
)  e.  Fin )
2928ralimi 2860 . . . . . . 7  |-  ( A. f  e.  M  f finSupp  Z  ->  A. f  e.  M  ( f supp  Z )  e.  Fin )
3029adantr 465 . . . . . 6  |-  ( ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )  ->  A. f  e.  M  ( f supp  Z )  e.  Fin )
31 iunfi 7820 . . . . . 6  |-  ( ( M  e.  Fin  /\  A. f  e.  M  ( f supp  Z )  e. 
Fin )  ->  U_ f  e.  M  ( f supp  Z )  e.  Fin )
3226, 30, 31syl2an 477 . . . . 5  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U_ f  e.  M  ( f supp  Z )  e.  Fin )
331, 32syl5eqel 2559 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U  e.  Fin )
34 simprr 756 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U  =/=  (/) )
358, 9, 103syl 20 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  dom  f  =  NN0 )
3635ex 434 . . . . . . . . . . . 12  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  dom  f  =  NN0 ) )
37363ad2ant1 1017 . . . . . . . . . . 11  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  ->  dom  f  =  NN0 ) )
3837adantr 465 . . . . . . . . . 10  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  (
f  e.  M  ->  dom  f  =  NN0 ) )
3938imp 429 . . . . . . . . 9  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  /\  f  e.  M )  ->  dom  f  =  NN0 )
40 nn0ssre 10811 . . . . . . . . 9  |-  NN0  C_  RR
4139, 40syl6eqss 3559 . . . . . . . 8  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  /\  f  e.  M )  ->  dom  f  C_  RR )
427, 41syl5ss 3520 . . . . . . 7  |-  ( ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  /\  f  e.  M )  ->  ( f supp  Z ) 
C_  RR )
4342ex 434 . . . . . 6  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  (
f  e.  M  -> 
( f supp  Z ) 
C_  RR ) )
446, 43ralrimi 2867 . . . . 5  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  A. f  e.  M  ( f supp  Z )  C_  RR )
451sseq1i 3533 . . . . . 6  |-  ( U 
C_  RR  <->  U_ f  e.  M  ( f supp  Z )  C_  RR )
46 iunss 4372 . . . . . 6  |-  ( U_ f  e.  M  (
f supp  Z )  C_  RR  <->  A. f  e.  M  ( f supp  Z )  C_  RR )
4745, 46bitri 249 . . . . 5  |-  ( U 
C_  RR  <->  A. f  e.  M  ( f supp  Z )  C_  RR )
4844, 47sylibr 212 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U  C_  RR )
49 fsuppmapnn0fiub.s . . . . 5  |-  S  =  sup ( U ,  RR ,  <  )
50 fisupcl 7939 . . . . 5  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  sup ( U ,  RR ,  <  )  e.  U
)
5149, 50syl5eqel 2559 . . . 4  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  S  e.  U )
5225, 33, 34, 48, 51syl13anc 1230 . . 3  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  S  e.  U )
5323, 52sseldd 3510 . 2  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  S  e.  NN0 )
5453ex 434 1  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )  ->  S  e.  NN0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    C_ wss 3481   (/)c0 3790   U_ciun 4331   class class class wbr 4453    Or wor 4805   dom cdm 5005    Fn wfn 5589  (class class class)co 6295   supp csupp 6913    ^m cmap 7432   Fincfn 7528   finSupp cfsupp 7841   supcsup 7912   RRcr 9503    < clt 9640   NN0cn0 10807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-nn 10549  df-n0 10808
This theorem is referenced by:  fsuppmapnn0fiub  12077  fsuppmapnn0fiubex  12078
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