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Theorem fsuppmapnn0fiublem 12214
Description: Lemma for fsuppmapnn0fiub 12215 and fsuppmapnn0fiubex 12216. (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 1756 . . . . . . 7  |-  F/ f ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )
3 nfra1 2808 . . . . . . . 8  |-  F/ f A. f  e.  M  f finSupp  Z
4 nfv 1756 . . . . . . . 8  |-  F/ f  U  =/=  (/)
53, 4nfan 1989 . . . . . . 7  |-  F/ f ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )
62, 5nfan 1989 . . . . . 6  |-  F/ f ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )
7 suppssdm 6944 . . . . . . . 8  |-  ( f supp 
Z )  C_  dom  f
8 ssel2 3465 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
9 elmapfn 7511 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
10 fndm 5699 . . . . . . . . . . . . . 14  |-  ( f  Fn  NN0  ->  dom  f  =  NN0 )
11 eqimss 3522 . . . . . . . . . . . . . 14  |-  ( dom  f  =  NN0  ->  dom  f  C_  NN0 )
1210, 11syl 17 . . . . . . . . . . . . 13  |-  ( f  Fn  NN0  ->  dom  f  C_ 
NN0 )
138, 9, 123syl 18 . . . . . . . . . . . 12  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  dom  f  C_  NN0 )
1413ex 436 . . . . . . . . . . 11  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  dom  f  C_ 
NN0 ) )
15143ad2ant1 1027 . . . . . . . . . 10  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  ->  dom  f  C_  NN0 )
)
1615adantr 467 . . . . . . . . 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 431 . . . . . . . 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 3481 . . . . . . 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 436 . . . . . 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 2827 . . . . 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 4346 . . . . 5  |-  ( U_ f  e.  M  (
f supp  Z )  C_  NN0  <->  A. f  e.  M  ( f supp  Z )  C_  NN0 )
2220, 21sylibr 216 . . . 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 3514 . . 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 9727 . . . . 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 1007 . . . . . 6  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  M  e.  Fin )
27 id 23 . . . . . . . . 9  |-  ( f finSupp  Z  ->  f finSupp  Z )
2827fsuppimpd 7905 . . . . . . . 8  |-  ( f finSupp  Z  ->  ( f supp  Z
)  e.  Fin )
2928ralimi 2820 . . . . . . 7  |-  ( A. f  e.  M  f finSupp  Z  ->  A. f  e.  M  ( f supp  Z )  e.  Fin )
3029adantr 467 . . . . . 6  |-  ( ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )  ->  A. f  e.  M  ( f supp  Z )  e.  Fin )
31 iunfi 7877 . . . . . 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 480 . . . . 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 2516 . . . 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 765 . . . 4  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  U  =/=  (/) )
358, 9, 103syl 18 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  dom  f  =  NN0 )
3635ex 436 . . . . . . . . . . . 12  |-  ( M 
C_  ( R  ^m  NN0 )  ->  ( f  e.  M  ->  dom  f  =  NN0 ) )
37363ad2ant1 1027 . . . . . . . . . . 11  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  ->  (
f  e.  M  ->  dom  f  =  NN0 ) )
3837adantr 467 . . . . . . . . . 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 431 . . . . . . . . 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 10886 . . . . . . . . 9  |-  NN0  C_  RR
4139, 40syl6eqss 3520 . . . . . . . 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 3481 . . . . . . 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 436 . . . . . 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 2827 . . . . 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 3494 . . . . . 6  |-  ( U 
C_  RR  <->  U_ f  e.  M  ( f supp  Z )  C_  RR )
46 iunss 4346 . . . . . 6  |-  ( U_ f  e.  M  (
f supp  Z )  C_  RR  <->  A. f  e.  M  ( f supp  Z )  C_  RR )
4745, 46bitri 253 . . . . 5  |-  ( U 
C_  RR  <->  A. f  e.  M  ( f supp  Z )  C_  RR )
4844, 47sylibr 216 . . . 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 8000 . . . . 5  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  sup ( U ,  RR ,  <  )  e.  U
)
5149, 50syl5eqel 2516 . . . 4  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  S  e.  U )
5225, 33, 34, 48, 51syl13anc 1267 . . 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 3471 . 2  |-  ( ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )  ->  S  e.  NN0 )
5453ex 436 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 371    /\ w3a 983    = wceq 1438    e. wcel 1873    =/= wne 2619   A.wral 2776    C_ wss 3442   (/)c0 3767   U_ciun 4305   class class class wbr 4429    Or wor 4779   dom cdm 4859    Fn wfn 5602  (class class class)co 6311   supp csupp 6931    ^m cmap 7489   Fincfn 7586   finSupp cfsupp 7898   supcsup 7969   RRcr 9551    < clt 9688   NN0cn0 10882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-resscn 9609  ax-1cn 9610  ax-icn 9611  ax-addcl 9612  ax-addrcl 9613  ax-mulcl 9614  ax-mulrcl 9615  ax-i2m1 9620  ax-1ne0 9621  ax-rnegex 9623  ax-rrecex 9624  ax-cnre 9625  ax-pre-lttri 9626  ax-pre-lttrn 9627
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-riota 6273  df-ov 6314  df-oprab 6315  df-mpt2 6316  df-om 6713  df-1st 6813  df-2nd 6814  df-supp 6932  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-1o 7199  df-oadd 7203  df-er 7380  df-map 7491  df-en 7587  df-dom 7588  df-sdom 7589  df-fin 7590  df-fsupp 7899  df-sup 7971  df-pnf 9690  df-mnf 9691  df-ltxr 9693  df-nn 10623  df-n0 10883
This theorem is referenced by:  fsuppmapnn0fiub  12215  fsuppmapnn0fiubex  12216
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