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Theorem strleund 12348
Description: Combine two structures into one. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 27-Jan-2023.)
Hypotheses
Ref Expression
strleund.f  |-  ( ph  ->  F Struct  <. A ,  B >. )
strleund.g  |-  ( ph  ->  G Struct  <. C ,  D >. )
strleund.l  |-  ( ph  ->  B  <  C )
Assertion
Ref Expression
strleund  |-  ( ph  ->  ( F  u.  G
) Struct  <. A ,  D >. )

Proof of Theorem strleund
StepHypRef Expression
1 strleund.f . . . . 5  |-  ( ph  ->  F Struct  <. A ,  B >. )
2 isstructim 12274 . . . . 5  |-  ( F Struct  <. A ,  B >.  -> 
( ( A  e.  NN  /\  B  e.  NN  /\  A  <_  B )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( A ... B
) ) )
31, 2syl 14 . . . 4  |-  ( ph  ->  ( ( A  e.  NN  /\  B  e.  NN  /\  A  <_  B )  /\  Fun  ( F  \  { (/) } )  /\  dom  F  C_  ( A ... B
) ) )
43simp1d 994 . . 3  |-  ( ph  ->  ( A  e.  NN  /\  B  e.  NN  /\  A  <_  B ) )
54simp1d 994 . 2  |-  ( ph  ->  A  e.  NN )
6 strleund.g . . . . 5  |-  ( ph  ->  G Struct  <. C ,  D >. )
7 isstructim 12274 . . . . 5  |-  ( G Struct  <. C ,  D >.  -> 
( ( C  e.  NN  /\  D  e.  NN  /\  C  <_  D )  /\  Fun  ( G  \  { (/) } )  /\  dom  G  C_  ( C ... D
) ) )
86, 7syl 14 . . . 4  |-  ( ph  ->  ( ( C  e.  NN  /\  D  e.  NN  /\  C  <_  D )  /\  Fun  ( G  \  { (/) } )  /\  dom  G  C_  ( C ... D
) ) )
98simp1d 994 . . 3  |-  ( ph  ->  ( C  e.  NN  /\  D  e.  NN  /\  C  <_  D ) )
109simp2d 995 . 2  |-  ( ph  ->  D  e.  NN )
115nnred 8851 . . 3  |-  ( ph  ->  A  e.  RR )
129simp1d 994 . . . 4  |-  ( ph  ->  C  e.  NN )
1312nnred 8851 . . 3  |-  ( ph  ->  C  e.  RR )
1410nnred 8851 . . 3  |-  ( ph  ->  D  e.  RR )
154simp2d 995 . . . . 5  |-  ( ph  ->  B  e.  NN )
1615nnred 8851 . . . 4  |-  ( ph  ->  B  e.  RR )
174simp3d 996 . . . 4  |-  ( ph  ->  A  <_  B )
18 strleund.l . . . . 5  |-  ( ph  ->  B  <  C )
1916, 13, 18ltled 7998 . . . 4  |-  ( ph  ->  B  <_  C )
2011, 16, 13, 17, 19letrd 8003 . . 3  |-  ( ph  ->  A  <_  C )
219simp3d 996 . . 3  |-  ( ph  ->  C  <_  D )
2211, 13, 14, 20, 21letrd 8003 . 2  |-  ( ph  ->  A  <_  D )
233simp2d 995 . . . 4  |-  ( ph  ->  Fun  ( F  \  { (/) } ) )
248simp2d 995 . . . 4  |-  ( ph  ->  Fun  ( G  \  { (/) } ) )
25 difss 3234 . . . . . . . 8  |-  ( F 
\  { (/) } ) 
C_  F
26 dmss 4787 . . . . . . . 8  |-  ( ( F  \  { (/) } )  C_  F  ->  dom  ( F  \  { (/)
} )  C_  dom  F )
2725, 26mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  dom  F )
283simp3d 996 . . . . . . 7  |-  ( ph  ->  dom  F  C_  ( A ... B ) )
2927, 28sstrd 3138 . . . . . 6  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  ( A ... B ) )
30 difss 3234 . . . . . . . 8  |-  ( G 
\  { (/) } ) 
C_  G
31 dmss 4787 . . . . . . . 8  |-  ( ( G  \  { (/) } )  C_  G  ->  dom  ( G  \  { (/)
} )  C_  dom  G )
3230, 31mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  dom  G )
338simp3d 996 . . . . . . 7  |-  ( ph  ->  dom  G  C_  ( C ... D ) )
3432, 33sstrd 3138 . . . . . 6  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  ( C ... D ) )
35 ss2in 3336 . . . . . 6  |-  ( ( dom  ( F  \  { (/) } )  C_  ( A ... B )  /\  dom  ( G 
\  { (/) } ) 
C_  ( C ... D ) )  -> 
( dom  ( F  \  { (/) } )  i^i 
dom  ( G  \  { (/) } ) ) 
C_  ( ( A ... B )  i^i  ( C ... D
) ) )
3629, 34, 35syl2anc 409 . . . . 5  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) ) 
C_  ( ( A ... B )  i^i  ( C ... D
) ) )
37 fzdisj 9960 . . . . . 6  |-  ( B  <  C  ->  (
( A ... B
)  i^i  ( C ... D ) )  =  (/) )
3818, 37syl 14 . . . . 5  |-  ( ph  ->  ( ( A ... B )  i^i  ( C ... D ) )  =  (/) )
39 sseq0 3436 . . . . 5  |-  ( ( ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) ) 
C_  ( ( A ... B )  i^i  ( C ... D
) )  /\  (
( A ... B
)  i^i  ( C ... D ) )  =  (/) )  ->  ( dom  ( F  \  { (/)
} )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )
4036, 38, 39syl2anc 409 . . . 4  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )
41 funun 5216 . . . 4  |-  ( ( ( Fun  ( F 
\  { (/) } )  /\  Fun  ( G 
\  { (/) } ) )  /\  ( dom  ( F  \  { (/)
} )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )  ->  Fun  ( ( F  \  { (/) } )  u.  ( G  \  { (/) } ) ) )
4223, 24, 40, 41syl21anc 1219 . . 3  |-  ( ph  ->  Fun  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) ) )
43 difundir 3361 . . . 4  |-  ( ( F  u.  G ) 
\  { (/) } )  =  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) )
4443funeqi 5193 . . 3  |-  ( Fun  ( ( F  u.  G )  \  { (/)
} )  <->  Fun  ( ( F  \  { (/) } )  u.  ( G 
\  { (/) } ) ) )
4542, 44sylibr 133 . 2  |-  ( ph  ->  Fun  ( ( F  u.  G )  \  { (/) } ) )
46 structex 12272 . . . 4  |-  ( F Struct  <. A ,  B >.  ->  F  e.  _V )
471, 46syl 14 . . 3  |-  ( ph  ->  F  e.  _V )
48 structex 12272 . . . 4  |-  ( G Struct  <. C ,  D >.  ->  G  e.  _V )
496, 48syl 14 . . 3  |-  ( ph  ->  G  e.  _V )
50 unexg 4405 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F  u.  G
)  e.  _V )
5147, 49, 50syl2anc 409 . 2  |-  ( ph  ->  ( F  u.  G
)  e.  _V )
52 dmun 4795 . . 3  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
5315nnzd 9290 . . . . . . 7  |-  ( ph  ->  B  e.  ZZ )
5410nnzd 9290 . . . . . . 7  |-  ( ph  ->  D  e.  ZZ )
5516, 13, 14, 19, 21letrd 8003 . . . . . . 7  |-  ( ph  ->  B  <_  D )
56 eluz2 9450 . . . . . . 7  |-  ( D  e.  ( ZZ>= `  B
)  <->  ( B  e.  ZZ  /\  D  e.  ZZ  /\  B  <_  D ) )
5753, 54, 55, 56syl3anbrc 1166 . . . . . 6  |-  ( ph  ->  D  e.  ( ZZ>= `  B ) )
58 fzss2 9972 . . . . . 6  |-  ( D  e.  ( ZZ>= `  B
)  ->  ( A ... B )  C_  ( A ... D ) )
5957, 58syl 14 . . . . 5  |-  ( ph  ->  ( A ... B
)  C_  ( A ... D ) )
6028, 59sstrd 3138 . . . 4  |-  ( ph  ->  dom  F  C_  ( A ... D ) )
615nnzd 9290 . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
6212nnzd 9290 . . . . . . 7  |-  ( ph  ->  C  e.  ZZ )
63 eluz2 9450 . . . . . . 7  |-  ( C  e.  ( ZZ>= `  A
)  <->  ( A  e.  ZZ  /\  C  e.  ZZ  /\  A  <_  C ) )
6461, 62, 20, 63syl3anbrc 1166 . . . . . 6  |-  ( ph  ->  C  e.  ( ZZ>= `  A ) )
65 fzss1 9971 . . . . . 6  |-  ( C  e.  ( ZZ>= `  A
)  ->  ( C ... D )  C_  ( A ... D ) )
6664, 65syl 14 . . . . 5  |-  ( ph  ->  ( C ... D
)  C_  ( A ... D ) )
6733, 66sstrd 3138 . . . 4  |-  ( ph  ->  dom  G  C_  ( A ... D ) )
6860, 67unssd 3284 . . 3  |-  ( ph  ->  ( dom  F  u.  dom  G )  C_  ( A ... D ) )
6952, 68eqsstrid 3174 . 2  |-  ( ph  ->  dom  ( F  u.  G )  C_  ( A ... D ) )
70 isstructr 12275 . 2  |-  ( ( ( A  e.  NN  /\  D  e.  NN  /\  A  <_  D )  /\  ( Fun  ( ( F  u.  G )  \  { (/) } )  /\  ( F  u.  G
)  e.  _V  /\  dom  ( F  u.  G
)  C_  ( A ... D ) ) )  ->  ( F  u.  G ) Struct  <. A ,  D >. )
715, 10, 22, 45, 51, 69, 70syl33anc 1235 1  |-  ( ph  ->  ( F  u.  G
) Struct  <. A ,  D >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    = wceq 1335    e. wcel 2128   _Vcvv 2712    \ cdif 3099    u. cun 3100    i^i cin 3101    C_ wss 3102   (/)c0 3395   {csn 3561   <.cop 3564   class class class wbr 3967   dom cdm 4588   Fun wfun 5166   ` cfv 5172  (class class class)co 5826    < clt 7914    <_ cle 7915   NNcn 8838   ZZcz 9172   ZZ>=cuz 9444   ...cfz 9918   Struct cstr 12256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-addcom 7834  ax-addass 7836  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-ltadd 7850
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-fz 9919  df-struct 12262
This theorem is referenced by:  strle2g  12351  strle3g  12352  srngstrd  12382  lmodstrd  12393  ipsstrd  12401
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