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Theorem strleund 11942
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 11868 . . . . 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 976 . . 3  |-  ( ph  ->  ( A  e.  NN  /\  B  e.  NN  /\  A  <_  B ) )
54simp1d 976 . 2  |-  ( ph  ->  A  e.  NN )
6 strleund.g . . . . 5  |-  ( ph  ->  G Struct  <. C ,  D >. )
7 isstructim 11868 . . . . 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 976 . . 3  |-  ( ph  ->  ( C  e.  NN  /\  D  e.  NN  /\  C  <_  D ) )
109simp2d 977 . 2  |-  ( ph  ->  D  e.  NN )
115nnred 8690 . . 3  |-  ( ph  ->  A  e.  RR )
129simp1d 976 . . . 4  |-  ( ph  ->  C  e.  NN )
1312nnred 8690 . . 3  |-  ( ph  ->  C  e.  RR )
1410nnred 8690 . . 3  |-  ( ph  ->  D  e.  RR )
154simp2d 977 . . . . 5  |-  ( ph  ->  B  e.  NN )
1615nnred 8690 . . . 4  |-  ( ph  ->  B  e.  RR )
174simp3d 978 . . . 4  |-  ( ph  ->  A  <_  B )
18 strleund.l . . . . 5  |-  ( ph  ->  B  <  C )
1916, 13, 18ltled 7845 . . . 4  |-  ( ph  ->  B  <_  C )
2011, 16, 13, 17, 19letrd 7850 . . 3  |-  ( ph  ->  A  <_  C )
219simp3d 978 . . 3  |-  ( ph  ->  C  <_  D )
2211, 13, 14, 20, 21letrd 7850 . 2  |-  ( ph  ->  A  <_  D )
233simp2d 977 . . . 4  |-  ( ph  ->  Fun  ( F  \  { (/) } ) )
248simp2d 977 . . . 4  |-  ( ph  ->  Fun  ( G  \  { (/) } ) )
25 difss 3170 . . . . . . . 8  |-  ( F 
\  { (/) } ) 
C_  F
26 dmss 4706 . . . . . . . 8  |-  ( ( F  \  { (/) } )  C_  F  ->  dom  ( F  \  { (/)
} )  C_  dom  F )
2725, 26mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  dom  F )
283simp3d 978 . . . . . . 7  |-  ( ph  ->  dom  F  C_  ( A ... B ) )
2927, 28sstrd 3075 . . . . . 6  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  ( A ... B ) )
30 difss 3170 . . . . . . . 8  |-  ( G 
\  { (/) } ) 
C_  G
31 dmss 4706 . . . . . . . 8  |-  ( ( G  \  { (/) } )  C_  G  ->  dom  ( G  \  { (/)
} )  C_  dom  G )
3230, 31mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  dom  G )
338simp3d 978 . . . . . . 7  |-  ( ph  ->  dom  G  C_  ( C ... D ) )
3432, 33sstrd 3075 . . . . . 6  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  ( C ... D ) )
35 ss2in 3272 . . . . . 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 406 . . . . 5  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) ) 
C_  ( ( A ... B )  i^i  ( C ... D
) ) )
37 fzdisj 9772 . . . . . 6  |-  ( B  <  C  ->  (
( A ... B
)  i^i  ( C ... D ) )  =  (/) )
3818, 37syl 14 . . . . 5  |-  ( ph  ->  ( ( A ... B )  i^i  ( C ... D ) )  =  (/) )
39 sseq0 3372 . . . . 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 406 . . . 4  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )
41 funun 5135 . . . 4  |-  ( ( ( Fun  ( F 
\  { (/) } )  /\  Fun  ( G 
\  { (/) } ) )  /\  ( dom  ( F  \  { (/)
} )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )  ->  Fun  ( ( F  \  { (/) } )  u.  ( G  \  { (/) } ) ) )
4223, 24, 40, 41syl21anc 1198 . . 3  |-  ( ph  ->  Fun  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) ) )
43 difundir 3297 . . . 4  |-  ( ( F  u.  G ) 
\  { (/) } )  =  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) )
4443funeqi 5112 . . 3  |-  ( Fun  ( ( F  u.  G )  \  { (/)
} )  <->  Fun  ( ( F  \  { (/) } )  u.  ( G 
\  { (/) } ) ) )
4542, 44sylibr 133 . 2  |-  ( ph  ->  Fun  ( ( F  u.  G )  \  { (/) } ) )
46 structex 11866 . . . 4  |-  ( F Struct  <. A ,  B >.  ->  F  e.  _V )
471, 46syl 14 . . 3  |-  ( ph  ->  F  e.  _V )
48 structex 11866 . . . 4  |-  ( G Struct  <. C ,  D >.  ->  G  e.  _V )
496, 48syl 14 . . 3  |-  ( ph  ->  G  e.  _V )
50 unexg 4332 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F  u.  G
)  e.  _V )
5147, 49, 50syl2anc 406 . 2  |-  ( ph  ->  ( F  u.  G
)  e.  _V )
52 dmun 4714 . . 3  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
5315nnzd 9123 . . . . . . 7  |-  ( ph  ->  B  e.  ZZ )
5410nnzd 9123 . . . . . . 7  |-  ( ph  ->  D  e.  ZZ )
5516, 13, 14, 19, 21letrd 7850 . . . . . . 7  |-  ( ph  ->  B  <_  D )
56 eluz2 9281 . . . . . . 7  |-  ( D  e.  ( ZZ>= `  B
)  <->  ( B  e.  ZZ  /\  D  e.  ZZ  /\  B  <_  D ) )
5753, 54, 55, 56syl3anbrc 1148 . . . . . 6  |-  ( ph  ->  D  e.  ( ZZ>= `  B ) )
58 fzss2 9784 . . . . . 6  |-  ( D  e.  ( ZZ>= `  B
)  ->  ( A ... B )  C_  ( A ... D ) )
5957, 58syl 14 . . . . 5  |-  ( ph  ->  ( A ... B
)  C_  ( A ... D ) )
6028, 59sstrd 3075 . . . 4  |-  ( ph  ->  dom  F  C_  ( A ... D ) )
615nnzd 9123 . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
6212nnzd 9123 . . . . . . 7  |-  ( ph  ->  C  e.  ZZ )
63 eluz2 9281 . . . . . . 7  |-  ( C  e.  ( ZZ>= `  A
)  <->  ( A  e.  ZZ  /\  C  e.  ZZ  /\  A  <_  C ) )
6461, 62, 20, 63syl3anbrc 1148 . . . . . 6  |-  ( ph  ->  C  e.  ( ZZ>= `  A ) )
65 fzss1 9783 . . . . . 6  |-  ( C  e.  ( ZZ>= `  A
)  ->  ( C ... D )  C_  ( A ... D ) )
6664, 65syl 14 . . . . 5  |-  ( ph  ->  ( C ... D
)  C_  ( A ... D ) )
6733, 66sstrd 3075 . . . 4  |-  ( ph  ->  dom  G  C_  ( A ... D ) )
6860, 67unssd 3220 . . 3  |-  ( ph  ->  ( dom  F  u.  dom  G )  C_  ( A ... D ) )
6952, 68eqsstrid 3111 . 2  |-  ( ph  ->  dom  ( F  u.  G )  C_  ( A ... D ) )
70 isstructr 11869 . 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 1214 1  |-  ( ph  ->  ( F  u.  G
) Struct  <. A ,  D >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 945    = wceq 1314    e. wcel 1463   _Vcvv 2658    \ cdif 3036    u. cun 3037    i^i cin 3038    C_ wss 3039   (/)c0 3331   {csn 3495   <.cop 3498   class class class wbr 3897   dom cdm 4507   Fun wfun 5085   ` cfv 5091  (class class class)co 5740    < clt 7764    <_ cle 7765   NNcn 8677   ZZcz 9005   ZZ>=cuz 9275   ...cfz 9730   Struct cstr 11850
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8678  df-n0 8929  df-z 9006  df-uz 9276  df-fz 9731  df-struct 11856
This theorem is referenced by:  strle2g  11945  strle3g  11946  srngstrd  11976  lmodstrd  11987  ipsstrd  11995
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