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Theorem strleund 11636
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 11562 . . . . 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 956 . . 3  |-  ( ph  ->  ( A  e.  NN  /\  B  e.  NN  /\  A  <_  B ) )
54simp1d 956 . 2  |-  ( ph  ->  A  e.  NN )
6 strleund.g . . . . 5  |-  ( ph  ->  G Struct  <. C ,  D >. )
7 isstructim 11562 . . . . 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 956 . . 3  |-  ( ph  ->  ( C  e.  NN  /\  D  e.  NN  /\  C  <_  D ) )
109simp2d 957 . 2  |-  ( ph  ->  D  e.  NN )
115nnred 8489 . . 3  |-  ( ph  ->  A  e.  RR )
129simp1d 956 . . . 4  |-  ( ph  ->  C  e.  NN )
1312nnred 8489 . . 3  |-  ( ph  ->  C  e.  RR )
1410nnred 8489 . . 3  |-  ( ph  ->  D  e.  RR )
154simp2d 957 . . . . 5  |-  ( ph  ->  B  e.  NN )
1615nnred 8489 . . . 4  |-  ( ph  ->  B  e.  RR )
174simp3d 958 . . . 4  |-  ( ph  ->  A  <_  B )
18 strleund.l . . . . 5  |-  ( ph  ->  B  <  C )
1916, 13, 18ltled 7656 . . . 4  |-  ( ph  ->  B  <_  C )
2011, 16, 13, 17, 19letrd 7661 . . 3  |-  ( ph  ->  A  <_  C )
219simp3d 958 . . 3  |-  ( ph  ->  C  <_  D )
2211, 13, 14, 20, 21letrd 7661 . 2  |-  ( ph  ->  A  <_  D )
233simp2d 957 . . . 4  |-  ( ph  ->  Fun  ( F  \  { (/) } ) )
248simp2d 957 . . . 4  |-  ( ph  ->  Fun  ( G  \  { (/) } ) )
25 difss 3127 . . . . . . . 8  |-  ( F 
\  { (/) } ) 
C_  F
26 dmss 4648 . . . . . . . 8  |-  ( ( F  \  { (/) } )  C_  F  ->  dom  ( F  \  { (/)
} )  C_  dom  F )
2725, 26mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  dom  F )
283simp3d 958 . . . . . . 7  |-  ( ph  ->  dom  F  C_  ( A ... B ) )
2927, 28sstrd 3036 . . . . . 6  |-  ( ph  ->  dom  ( F  \  { (/) } )  C_  ( A ... B ) )
30 difss 3127 . . . . . . . 8  |-  ( G 
\  { (/) } ) 
C_  G
31 dmss 4648 . . . . . . . 8  |-  ( ( G  \  { (/) } )  C_  G  ->  dom  ( G  \  { (/)
} )  C_  dom  G )
3230, 31mp1i 10 . . . . . . 7  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  dom  G )
338simp3d 958 . . . . . . 7  |-  ( ph  ->  dom  G  C_  ( C ... D ) )
3432, 33sstrd 3036 . . . . . 6  |-  ( ph  ->  dom  ( G  \  { (/) } )  C_  ( C ... D ) )
35 ss2in 3228 . . . . . 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 404 . . . . 5  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) ) 
C_  ( ( A ... B )  i^i  ( C ... D
) ) )
37 fzdisj 9520 . . . . . 6  |-  ( B  <  C  ->  (
( A ... B
)  i^i  ( C ... D ) )  =  (/) )
3818, 37syl 14 . . . . 5  |-  ( ph  ->  ( ( A ... B )  i^i  ( C ... D ) )  =  (/) )
39 sseq0 3328 . . . . 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 404 . . . 4  |-  ( ph  ->  ( dom  ( F 
\  { (/) } )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )
41 funun 5071 . . . 4  |-  ( ( ( Fun  ( F 
\  { (/) } )  /\  Fun  ( G 
\  { (/) } ) )  /\  ( dom  ( F  \  { (/)
} )  i^i  dom  ( G  \  { (/) } ) )  =  (/) )  ->  Fun  ( ( F  \  { (/) } )  u.  ( G  \  { (/) } ) ) )
4223, 24, 40, 41syl21anc 1174 . . 3  |-  ( ph  ->  Fun  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) ) )
43 difundir 3253 . . . 4  |-  ( ( F  u.  G ) 
\  { (/) } )  =  ( ( F 
\  { (/) } )  u.  ( G  \  { (/) } ) )
4443funeqi 5049 . . 3  |-  ( Fun  ( ( F  u.  G )  \  { (/)
} )  <->  Fun  ( ( F  \  { (/) } )  u.  ( G 
\  { (/) } ) ) )
4542, 44sylibr 133 . 2  |-  ( ph  ->  Fun  ( ( F  u.  G )  \  { (/) } ) )
46 structex 11560 . . . 4  |-  ( F Struct  <. A ,  B >.  ->  F  e.  _V )
471, 46syl 14 . . 3  |-  ( ph  ->  F  e.  _V )
48 structex 11560 . . . 4  |-  ( G Struct  <. C ,  D >.  ->  G  e.  _V )
496, 48syl 14 . . 3  |-  ( ph  ->  G  e.  _V )
50 unexg 4278 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V )  ->  ( F  u.  G
)  e.  _V )
5147, 49, 50syl2anc 404 . 2  |-  ( ph  ->  ( F  u.  G
)  e.  _V )
52 dmun 4656 . . 3  |-  dom  ( F  u.  G )  =  ( dom  F  u.  dom  G )
5315nnzd 8921 . . . . . . 7  |-  ( ph  ->  B  e.  ZZ )
5410nnzd 8921 . . . . . . 7  |-  ( ph  ->  D  e.  ZZ )
5516, 13, 14, 19, 21letrd 7661 . . . . . . 7  |-  ( ph  ->  B  <_  D )
56 eluz2 9079 . . . . . . 7  |-  ( D  e.  ( ZZ>= `  B
)  <->  ( B  e.  ZZ  /\  D  e.  ZZ  /\  B  <_  D ) )
5753, 54, 55, 56syl3anbrc 1128 . . . . . 6  |-  ( ph  ->  D  e.  ( ZZ>= `  B ) )
58 fzss2 9532 . . . . . 6  |-  ( D  e.  ( ZZ>= `  B
)  ->  ( A ... B )  C_  ( A ... D ) )
5957, 58syl 14 . . . . 5  |-  ( ph  ->  ( A ... B
)  C_  ( A ... D ) )
6028, 59sstrd 3036 . . . 4  |-  ( ph  ->  dom  F  C_  ( A ... D ) )
615nnzd 8921 . . . . . . 7  |-  ( ph  ->  A  e.  ZZ )
6212nnzd 8921 . . . . . . 7  |-  ( ph  ->  C  e.  ZZ )
63 eluz2 9079 . . . . . . 7  |-  ( C  e.  ( ZZ>= `  A
)  <->  ( A  e.  ZZ  /\  C  e.  ZZ  /\  A  <_  C ) )
6461, 62, 20, 63syl3anbrc 1128 . . . . . 6  |-  ( ph  ->  C  e.  ( ZZ>= `  A ) )
65 fzss1 9531 . . . . . 6  |-  ( C  e.  ( ZZ>= `  A
)  ->  ( C ... D )  C_  ( A ... D ) )
6664, 65syl 14 . . . . 5  |-  ( ph  ->  ( C ... D
)  C_  ( A ... D ) )
6733, 66sstrd 3036 . . . 4  |-  ( ph  ->  dom  G  C_  ( A ... D ) )
6860, 67unssd 3177 . . 3  |-  ( ph  ->  ( dom  F  u.  dom  G )  C_  ( A ... D ) )
6952, 68syl5eqss 3071 . 2  |-  ( ph  ->  dom  ( F  u.  G )  C_  ( A ... D ) )
70 isstructr 11563 . 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 1190 1  |-  ( ph  ->  ( F  u.  G
) Struct  <. A ,  D >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 925    = wceq 1290    e. wcel 1439   _Vcvv 2620    \ cdif 2997    u. cun 2998    i^i cin 2999    C_ wss 3000   (/)c0 3287   {csn 3450   <.cop 3453   class class class wbr 3851   dom cdm 4451   Fun wfun 5022   ` cfv 5028  (class class class)co 5666    < clt 7576    <_ cle 7577   NNcn 8476   ZZcz 8804   ZZ>=cuz 9073   ...cfz 9478   Struct cstr 11544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-addcom 7499  ax-addass 7501  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-0id 7507  ax-rnegex 7508  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-ltadd 7515
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-inn 8477  df-n0 8728  df-z 8805  df-uz 9074  df-fz 9479  df-struct 11550
This theorem is referenced by:  strle2g  11639  strle3g  11640  srngstrd  11670  lmodstrd  11681  ipsstrd  11689
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