Theorem strext 13405

Description: Extending the upper range of a structure. This works because when we say that a structure has components in A ... C we are not saying that every slot in that range is present, just that all the slots that are present are within that range. (Contributed by Jim Kingdon, 26-Feb-2025.)

Hypotheses

Ref	Expression
strext.f	|- ( ph -> F Struct <. A , B >. )
strext.c	|- ( ph -> C e. ( ZZ>= ` B ) )

Assertion

Ref	Expression
strext	|- ( ph -> F Struct <. A , C >. )

Proof of Theorem strext

Step	Hyp	Ref	Expression
1		strext.f	. . . . 5 |- ( ph -> F Struct <. A , B >. )
2		isstructim 13313	. . . . 5 |- ( F Struct <. A , B >. -> ( ( A e. NN /\ B e. NN /\ A <_ B ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( A ... B ) ) )
3	1, 2	syl 14	. . . 4 |- ( ph -> ( ( A e. NN /\ B e. NN /\ A <_ B ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( A ... B ) ) )
4	3	simp1d 1036	. . 3 |- ( ph -> ( A e. NN /\ B e. NN /\ A <_ B ) )
5	4	simp1d 1036	. 2 |- ( ph -> A e. NN )
6	4	simp2d 1037	. . 3 |- ( ph -> B e. NN )
7		strext.c	. . 3 |- ( ph -> C e. ( ZZ>= ` B ) )
8		eluznn 9953	. . 3 |- ( ( B e. NN /\ C e. ( ZZ>= ` B ) ) -> C e. NN )
9	6, 7, 8	syl2anc 411	. 2 |- ( ph -> C e. NN )
10	5	nnred 9270	. . 3 |- ( ph -> A e. RR )
11	6	nnred 9270	. . 3 |- ( ph -> B e. RR )
12	9	nnred 9270	. . 3 |- ( ph -> C e. RR )
13	4	simp3d 1038	. . 3 |- ( ph -> A <_ B )
14		eluzle 9887	. . . 4 |- ( C e. ( ZZ>= ` B ) -> B <_ C )
15	7, 14	syl 14	. . 3 |- ( ph -> B <_ C )
16	10, 11, 12, 13, 15	letrd 8414	. 2 |- ( ph -> A <_ C )
17	3	simp2d 1037	. 2 |- ( ph -> Fun ( F \ { (/) } ) )
18		structex 13311	. . 3 |- ( F Struct <. A , B >. -> F e. _V )
19	1, 18	syl 14	. 2 |- ( ph -> F e. _V )
20	3	simp3d 1038	. . 3 |- ( ph -> dom F C_ ( A ... B ) )
21		fzss2 10422	. . . 4 |- ( C e. ( ZZ>= ` B ) -> ( A ... B ) C_ ( A ... C ) )
22	7, 21	syl 14	. . 3 |- ( ph -> ( A ... B ) C_ ( A ... C ) )
23	20, 22	sstrd 3252	. 2 |- ( ph -> dom F C_ ( A ... C ) )
24		isstructr 13314	. 2 |- ( ( ( A e. NN /\ C e. NN /\ A <_ C ) /\ ( Fun ( F \ { (/) } ) /\ F e. _V /\ dom F C_ ( A ... C ) ) ) -> F Struct <. A , C >. )
25	5, 9, 16, 17, 19, 23, 24	syl33anc 1289	1 |- ( ph -> F Struct <. A , C >. )

Syntax hints:   -> wi 4   /\ w3a 1005   e. wcel 2205   _Vcvv 2815   \ cdif 3211   C_ wss 3214   (/)c0 3512   {csn 3694   <.cop 3697   class class class wbr 4114   dom cdm 4754   Fun wfun 5351   ` cfv 5357  (class class class)co 6058   <_ cle 8325   NNcn 9257   ZZ>=cuz 9874   ...cfz 10364   Struct cstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-inn 9258  df-z 9598  df-uz 9875  df-fz 10365  df-struct 13301

This theorem is referenced by: (None)