Theorem strext 12970

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 12879	. . . . 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 1012	. . 3 |- ( ph -> ( A e. NN /\ B e. NN /\ A <_ B ) )
5	4	simp1d 1012	. 2 |- ( ph -> A e. NN )
6	4	simp2d 1013	. . 3 |- ( ph -> B e. NN )
7		strext.c	. . 3 |- ( ph -> C e. ( ZZ>= ` B ) )
8		eluznn 9723	. . 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 9051	. . 3 |- ( ph -> A e. RR )
11	6	nnred 9051	. . 3 |- ( ph -> B e. RR )
12	9	nnred 9051	. . 3 |- ( ph -> C e. RR )
13	4	simp3d 1014	. . 3 |- ( ph -> A <_ B )
14		eluzle 9662	. . . 4 |- ( C e. ( ZZ>= ` B ) -> B <_ C )
15	7, 14	syl 14	. . 3 |- ( ph -> B <_ C )
16	10, 11, 12, 13, 15	letrd 8198	. 2 |- ( ph -> A <_ C )
17	3	simp2d 1013	. 2 |- ( ph -> Fun ( F \ { (/) } ) )
18		structex 12877	. . 3 |- ( F Struct <. A , B >. -> F e. _V )
19	1, 18	syl 14	. 2 |- ( ph -> F e. _V )
20	3	simp3d 1014	. . 3 |- ( ph -> dom F C_ ( A ... B ) )
21		fzss2 10188	. . . 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 3203	. 2 |- ( ph -> dom F C_ ( A ... C ) )
24		isstructr 12880	. 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 1265	1 |- ( ph -> F Struct <. A , C >. )

Syntax hints:   -> wi 4   /\ w3a 981   e. wcel 2176   _Vcvv 2772   \ cdif 3163   C_ wss 3166   (/)c0 3460   {csn 3633   <.cop 3636   class class class wbr 4045   dom cdm 4676   Fun wfun 5266   ` cfv 5272  (class class class)co 5946   <_ cle 8110   NNcn 9038   ZZ>=cuz 9650   ...cfz 10132   Struct cstr 12861

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-z 9375  df-uz 9651  df-fz 10133  df-struct 12867

This theorem is referenced by: (None)