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Theorem fsuppmapnn0fiublem 12098
Description: Lemma for fsuppmapnn0fiub 12099 and fsuppmapnn0fiubex 12100. (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 1708 . . . . . . 7  |-  F/ f ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )
3 nfra1 2838 . . . . . . . 8  |-  F/ f A. f  e.  M  f finSupp  Z
4 nfv 1708 . . . . . . . 8  |-  F/ f  U  =/=  (/)
53, 4nfan 1929 . . . . . . 7  |-  F/ f ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )
62, 5nfan 1929 . . . . . 6  |-  F/ f ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )
7 suppssdm 6930 . . . . . . . 8  |-  ( f supp 
Z )  C_  dom  f
8 ssel2 3494 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
9 elmapfn 7460 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
10 fndm 5686 . . . . . . . . . . . . . 14  |-  ( f  Fn  NN0  ->  dom  f  =  NN0 )
11 eqimss 3551 . . . . . . . . . . . . . 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 3510 . . . . . . 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 2857 . . . . 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 4373 . . . . 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 3543 . . 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 9682 . . . . 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 7854 . . . . . . . 8  |-  ( f finSupp  Z  ->  ( f supp  Z
)  e.  Fin )
2928ralimi 2850 . . . . . . 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 7826 . . . . . 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 2549 . . . 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 757 . . . 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 10820 . . . . . . . . 9  |-  NN0  C_  RR
4139, 40syl6eqss 3549 . . . . . . . 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 3510 . . . . . . 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 2857 . . . . 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 3523 . . . . . 6  |-  ( U 
C_  RR  <->  U_ f  e.  M  ( f supp  Z )  C_  RR )
46 iunss 4373 . . . . . 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 7945 . . . . 5  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  sup ( U ,  RR ,  <  )  e.  U
)
5149, 50syl5eqel 2549 . . . 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 3500 . 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 1395    e. wcel 1819    =/= wne 2652   A.wral 2807    C_ wss 3471   (/)c0 3793   U_ciun 4332   class class class wbr 4456    Or wor 4808   dom cdm 5008    Fn wfn 5589  (class class class)co 6296   supp csupp 6917    ^m cmap 7438   Fincfn 7535   finSupp cfsupp 7847   supcsup 7918   RRcr 9508    < clt 9645   NN0cn0 10816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-nn 10557  df-n0 10817
This theorem is referenced by:  fsuppmapnn0fiub  12099  fsuppmapnn0fiubex  12100
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