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Theorem fsuppmapnn0fiublem 30969
Description: Lemma for fsuppmapnn0fiub 30970 and fsuppmapnn0fiubex 30971. (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 1674 . . . . . . 7  |-  F/ f ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )
3 nfra1 2810 . . . . . . . 8  |-  F/ f A. f  e.  M  f finSupp  Z
4 nfv 1674 . . . . . . . 8  |-  F/ f  U  =/=  (/)
53, 4nfan 1866 . . . . . . 7  |-  F/ f ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) )
62, 5nfan 1866 . . . . . 6  |-  F/ f ( ( M  C_  ( R  ^m  NN0 )  /\  M  e.  Fin  /\  Z  e.  V )  /\  ( A. f  e.  M  f finSupp  Z  /\  U  =/=  (/) ) )
7 suppssdm 6816 . . . . . . . 8  |-  ( f supp 
Z )  C_  dom  f
8 ssel2 3462 . . . . . . . . . . . . 13  |-  ( ( M  C_  ( R  ^m  NN0 )  /\  f  e.  M )  ->  f  e.  ( R  ^m  NN0 ) )
9 elmapfn 7348 . . . . . . . . . . . . 13  |-  ( f  e.  ( R  ^m  NN0 )  ->  f  Fn  NN0 )
10 fndm 5621 . . . . . . . . . . . . . 14  |-  ( f  Fn  NN0  ->  dom  f  =  NN0 )
11 eqimss 3519 . . . . . . . . . . . . . 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 1009 . . . . . . . . . 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 3478 . . . . . . 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 2823 . . . . 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 4322 . . . . 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 3511 . . 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 9570 . . . . 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 989 . . . . . 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 7741 . . . . . . . 8  |-  ( f finSupp  Z  ->  ( f supp  Z
)  e.  Fin )
2928ralimi 2819 . . . . . . 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 7713 . . . . . 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 2546 . . . 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 1009 . . . . . . . . . . 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 10698 . . . . . . . . 9  |-  NN0  C_  RR
4139, 40syl6eqss 3517 . . . . . . . 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 3478 . . . . . . 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 2823 . . . . 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 3491 . . . . . 6  |-  ( U 
C_  RR  <->  U_ f  e.  M  ( f supp  Z )  C_  RR )
46 iunss 4322 . . . . . 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 7832 . . . . 5  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  sup ( U ,  RR ,  <  )  e.  U
)
5149, 50syl5eqel 2546 . . . 4  |-  ( (  <  Or  RR  /\  ( U  e.  Fin  /\  U  =/=  (/)  /\  U  C_  RR ) )  ->  S  e.  U )
5225, 33, 34, 48, 51syl13anc 1221 . . 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 3468 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799    C_ wss 3439   (/)c0 3748   U_ciun 4282   class class class wbr 4403    Or wor 4751   dom cdm 4951    Fn wfn 5524  (class class class)co 6203   supp csupp 6803    ^m cmap 7327   Fincfn 7423   finSupp cfsupp 7734   supcsup 7805   RRcr 9396    < clt 9533   NN0cn0 10694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-i2m1 9465  ax-1ne0 9466  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-sup 7806  df-pnf 9535  df-mnf 9536  df-ltxr 9538  df-nn 10438  df-n0 10695
This theorem is referenced by:  fsuppmapnn0fiub  30970  fsuppmapnn0fiubex  30971
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