Theorem strleund 13131

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

Step	Hyp	Ref	Expression
1		strleund.f	. . . . 5 |- ( ph -> F Struct <. A , B >. )
2		isstructim 13041	. . . . 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 1033	. . 3 |- ( ph -> ( A e. NN /\ B e. NN /\ A <_ B ) )
5	4	simp1d 1033	. 2 |- ( ph -> A e. NN )
6		strleund.g	. . . . 5 |- ( ph -> G Struct <. C , D >. )
7		isstructim 13041	. . . . 5 |- ( G Struct <. C , D >. -> ( ( C e. NN /\ D e. NN /\ C <_ D ) /\ Fun ( G \ { (/) } ) /\ dom G C_ ( C ... D ) ) )
8	6, 7	syl 14	. . . 4 |- ( ph -> ( ( C e. NN /\ D e. NN /\ C <_ D ) /\ Fun ( G \ { (/) } ) /\ dom G C_ ( C ... D ) ) )
9	8	simp1d 1033	. . 3 |- ( ph -> ( C e. NN /\ D e. NN /\ C <_ D ) )
10	9	simp2d 1034	. 2 |- ( ph -> D e. NN )
11	5	nnred 9119	. . 3 |- ( ph -> A e. RR )
12	9	simp1d 1033	. . . 4 |- ( ph -> C e. NN )
13	12	nnred 9119	. . 3 |- ( ph -> C e. RR )
14	10	nnred 9119	. . 3 |- ( ph -> D e. RR )
15	4	simp2d 1034	. . . . 5 |- ( ph -> B e. NN )
16	15	nnred 9119	. . . 4 |- ( ph -> B e. RR )
17	4	simp3d 1035	. . . 4 |- ( ph -> A <_ B )
18		strleund.l	. . . . 5 |- ( ph -> B < C )
19	16, 13, 18	ltled 8261	. . . 4 |- ( ph -> B <_ C )
20	11, 16, 13, 17, 19	letrd 8266	. . 3 |- ( ph -> A <_ C )
21	9	simp3d 1035	. . 3 |- ( ph -> C <_ D )
22	11, 13, 14, 20, 21	letrd 8266	. 2 |- ( ph -> A <_ D )
23	3	simp2d 1034	. . . 4 |- ( ph -> Fun ( F \ { (/) } ) )
24	8	simp2d 1034	. . . 4 |- ( ph -> Fun ( G \ { (/) } ) )
25		difss 3330	. . . . . . . 8 |- ( F \ { (/) } ) C_ F
26		dmss 4921	. . . . . . . 8 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
27	25, 26	mp1i 10	. . . . . . 7 |- ( ph -> dom ( F \ { (/) } ) C_ dom F )
28	3	simp3d 1035	. . . . . . 7 |- ( ph -> dom F C_ ( A ... B ) )
29	27, 28	sstrd 3234	. . . . . 6 |- ( ph -> dom ( F \ { (/) } ) C_ ( A ... B ) )
30		difss 3330	. . . . . . . 8 |- ( G \ { (/) } ) C_ G
31		dmss 4921	. . . . . . . 8 |- ( ( G \ { (/) } ) C_ G -> dom ( G \ { (/) } ) C_ dom G )
32	30, 31	mp1i 10	. . . . . . 7 |- ( ph -> dom ( G \ { (/) } ) C_ dom G )
33	8	simp3d 1035	. . . . . . 7 |- ( ph -> dom G C_ ( C ... D ) )
34	32, 33	sstrd 3234	. . . . . 6 |- ( ph -> dom ( G \ { (/) } ) C_ ( C ... D ) )
35		ss2in 3432	. . . . . 6 |- ( ( dom ( F \ { (/) } ) C_ ( A ... B ) /\ dom ( G \ { (/) } ) C_ ( C ... D ) ) -> ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) C_ ( ( A ... B ) i^i ( C ... D ) ) )
36	29, 34, 35	syl2anc 411	. . . . 5 |- ( ph -> ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) C_ ( ( A ... B ) i^i ( C ... D ) ) )
37		fzdisj 10244	. . . . . 6 |- ( B < C -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
38	18, 37	syl 14	. . . . 5 |- ( ph -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
39		sseq0 3533	. . . . 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 \ { (/) } ) ) = (/) )
40	36, 38, 39	syl2anc 411	. . . 4 |- ( ph -> ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) )
41		funun 5361	. . . 4 |- ( ( ( Fun ( F \ { (/) } ) /\ Fun ( G \ { (/) } ) ) /\ ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) ) -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
42	23, 24, 40, 41	syl21anc 1270	. . 3 |- ( ph -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
43		difundir 3457	. . . 4 |- ( ( F u. G ) \ { (/) } ) = ( ( F \ { (/) } ) u. ( G \ { (/) } ) )
44	43	funeqi 5338	. . 3 |- ( Fun ( ( F u. G ) \ { (/) } ) <-> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
45	42, 44	sylibr 134	. 2 |- ( ph -> Fun ( ( F u. G ) \ { (/) } ) )
46		structex 13039	. . . 4 |- ( F Struct <. A , B >. -> F e. _V )
47	1, 46	syl 14	. . 3 |- ( ph -> F e. _V )
48		structex 13039	. . . 4 |- ( G Struct <. C , D >. -> G e. _V )
49	6, 48	syl 14	. . 3 |- ( ph -> G e. _V )
50		unexg 4533	. . 3 |- ( ( F e. _V /\ G e. _V ) -> ( F u. G ) e. _V )
51	47, 49, 50	syl2anc 411	. 2 |- ( ph -> ( F u. G ) e. _V )
52		dmun 4929	. . 3 |- dom ( F u. G ) = ( dom F u. dom G )
53	15	nnzd 9564	. . . . . . 7 |- ( ph -> B e. ZZ )
54	10	nnzd 9564	. . . . . . 7 |- ( ph -> D e. ZZ )
55	16, 13, 14, 19, 21	letrd 8266	. . . . . . 7 |- ( ph -> B <_ D )
56		eluz2 9724	. . . . . . 7 |- ( D e. ( ZZ>= ` B ) <-> ( B e. ZZ /\ D e. ZZ /\ B <_ D ) )
57	53, 54, 55, 56	syl3anbrc 1205	. . . . . 6 |- ( ph -> D e. ( ZZ>= ` B ) )
58		fzss2 10256	. . . . . 6 |- ( D e. ( ZZ>= ` B ) -> ( A ... B ) C_ ( A ... D ) )
59	57, 58	syl 14	. . . . 5 |- ( ph -> ( A ... B ) C_ ( A ... D ) )
60	28, 59	sstrd 3234	. . . 4 |- ( ph -> dom F C_ ( A ... D ) )
61	5	nnzd 9564	. . . . . . 7 |- ( ph -> A e. ZZ )
62	12	nnzd 9564	. . . . . . 7 |- ( ph -> C e. ZZ )
63		eluz2 9724	. . . . . . 7 |- ( C e. ( ZZ>= ` A ) <-> ( A e. ZZ /\ C e. ZZ /\ A <_ C ) )
64	61, 62, 20, 63	syl3anbrc 1205	. . . . . 6 |- ( ph -> C e. ( ZZ>= ` A ) )
65		fzss1 10255	. . . . . 6 |- ( C e. ( ZZ>= ` A ) -> ( C ... D ) C_ ( A ... D ) )
66	64, 65	syl 14	. . . . 5 |- ( ph -> ( C ... D ) C_ ( A ... D ) )
67	33, 66	sstrd 3234	. . . 4 |- ( ph -> dom G C_ ( A ... D ) )
68	60, 67	unssd 3380	. . 3 |- ( ph -> ( dom F u. dom G ) C_ ( A ... D ) )
69	52, 68	eqsstrid 3270	. 2 |- ( ph -> dom ( F u. G ) C_ ( A ... D ) )
70		isstructr 13042	. 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 >. )
71	5, 10, 22, 45, 51, 69, 70	syl33anc 1286	1 |- ( ph -> ( F u. G ) Struct <. A , D >. )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   \ cdif 3194   u. cun 3195   i^i cin 3196   C_ wss 3197   (/)c0 3491   {csn 3666   <.cop 3669   class class class wbr 4082   dom cdm 4718   Fun wfun 5311   ` cfv 5317  (class class class)co 6000   < clt 8177   <_ cle 8178   NNcn 9106   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200   Struct cstr 13023

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-struct 13029

This theorem is referenced by:  strle2g  13135  strle3g  13136  srngstrd  13174  lmodstrd  13192  ipsstrd  13204  imasvalstrd  13298  prdsvalstrd  13299  psrvalstrd  14626