Theorem strleund 12483

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 12408	. . . . 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 999	. . 3 |- ( ph -> ( A e. NN /\ B e. NN /\ A <_ B ) )
5	4	simp1d 999	. 2 |- ( ph -> A e. NN )
6		strleund.g	. . . . 5 |- ( ph -> G Struct <. C , D >. )
7		isstructim 12408	. . . . 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 999	. . 3 |- ( ph -> ( C e. NN /\ D e. NN /\ C <_ D ) )
10	9	simp2d 1000	. 2 |- ( ph -> D e. NN )
11	5	nnred 8870	. . 3 |- ( ph -> A e. RR )
12	9	simp1d 999	. . . 4 |- ( ph -> C e. NN )
13	12	nnred 8870	. . 3 |- ( ph -> C e. RR )
14	10	nnred 8870	. . 3 |- ( ph -> D e. RR )
15	4	simp2d 1000	. . . . 5 |- ( ph -> B e. NN )
16	15	nnred 8870	. . . 4 |- ( ph -> B e. RR )
17	4	simp3d 1001	. . . 4 |- ( ph -> A <_ B )
18		strleund.l	. . . . 5 |- ( ph -> B < C )
19	16, 13, 18	ltled 8017	. . . 4 |- ( ph -> B <_ C )
20	11, 16, 13, 17, 19	letrd 8022	. . 3 |- ( ph -> A <_ C )
21	9	simp3d 1001	. . 3 |- ( ph -> C <_ D )
22	11, 13, 14, 20, 21	letrd 8022	. 2 |- ( ph -> A <_ D )
23	3	simp2d 1000	. . . 4 |- ( ph -> Fun ( F \ { (/) } ) )
24	8	simp2d 1000	. . . 4 |- ( ph -> Fun ( G \ { (/) } ) )
25		difss 3248	. . . . . . . 8 |- ( F \ { (/) } ) C_ F
26		dmss 4803	. . . . . . . 8 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
27	25, 26	mp1i 10	. . . . . . 7 |- ( ph -> dom ( F \ { (/) } ) C_ dom F )
28	3	simp3d 1001	. . . . . . 7 |- ( ph -> dom F C_ ( A ... B ) )
29	27, 28	sstrd 3152	. . . . . 6 |- ( ph -> dom ( F \ { (/) } ) C_ ( A ... B ) )
30		difss 3248	. . . . . . . 8 |- ( G \ { (/) } ) C_ G
31		dmss 4803	. . . . . . . 8 |- ( ( G \ { (/) } ) C_ G -> dom ( G \ { (/) } ) C_ dom G )
32	30, 31	mp1i 10	. . . . . . 7 |- ( ph -> dom ( G \ { (/) } ) C_ dom G )
33	8	simp3d 1001	. . . . . . 7 |- ( ph -> dom G C_ ( C ... D ) )
34	32, 33	sstrd 3152	. . . . . 6 |- ( ph -> dom ( G \ { (/) } ) C_ ( C ... D ) )
35		ss2in 3350	. . . . . 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 409	. . . . 5 |- ( ph -> ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) C_ ( ( A ... B ) i^i ( C ... D ) ) )
37		fzdisj 9987	. . . . . 6 |- ( B < C -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
38	18, 37	syl 14	. . . . 5 |- ( ph -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
39		sseq0 3450	. . . . 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 409	. . . 4 |- ( ph -> ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) )
41		funun 5232	. . . 4 |- ( ( ( Fun ( F \ { (/) } ) /\ Fun ( G \ { (/) } ) ) /\ ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) ) -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
42	23, 24, 40, 41	syl21anc 1227	. . 3 |- ( ph -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
43		difundir 3375	. . . 4 |- ( ( F u. G ) \ { (/) } ) = ( ( F \ { (/) } ) u. ( G \ { (/) } ) )
44	43	funeqi 5209	. . 3 |- ( Fun ( ( F u. G ) \ { (/) } ) <-> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
45	42, 44	sylibr 133	. 2 |- ( ph -> Fun ( ( F u. G ) \ { (/) } ) )
46		structex 12406	. . . 4 |- ( F Struct <. A , B >. -> F e. _V )
47	1, 46	syl 14	. . 3 |- ( ph -> F e. _V )
48		structex 12406	. . . 4 |- ( G Struct <. C , D >. -> G e. _V )
49	6, 48	syl 14	. . 3 |- ( ph -> G e. _V )
50		unexg 4421	. . 3 |- ( ( F e. _V /\ G e. _V ) -> ( F u. G ) e. _V )
51	47, 49, 50	syl2anc 409	. 2 |- ( ph -> ( F u. G ) e. _V )
52		dmun 4811	. . 3 |- dom ( F u. G ) = ( dom F u. dom G )
53	15	nnzd 9312	. . . . . . 7 |- ( ph -> B e. ZZ )
54	10	nnzd 9312	. . . . . . 7 |- ( ph -> D e. ZZ )
55	16, 13, 14, 19, 21	letrd 8022	. . . . . . 7 |- ( ph -> B <_ D )
56		eluz2 9472	. . . . . . 7 |- ( D e. ( ZZ>= ` B ) <-> ( B e. ZZ /\ D e. ZZ /\ B <_ D ) )
57	53, 54, 55, 56	syl3anbrc 1171	. . . . . 6 |- ( ph -> D e. ( ZZ>= ` B ) )
58		fzss2 9999	. . . . . 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 3152	. . . 4 |- ( ph -> dom F C_ ( A ... D ) )
61	5	nnzd 9312	. . . . . . 7 |- ( ph -> A e. ZZ )
62	12	nnzd 9312	. . . . . . 7 |- ( ph -> C e. ZZ )
63		eluz2 9472	. . . . . . 7 |- ( C e. ( ZZ>= ` A ) <-> ( A e. ZZ /\ C e. ZZ /\ A <_ C ) )
64	61, 62, 20, 63	syl3anbrc 1171	. . . . . 6 |- ( ph -> C e. ( ZZ>= ` A ) )
65		fzss1 9998	. . . . . 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 3152	. . . 4 |- ( ph -> dom G C_ ( A ... D ) )
68	60, 67	unssd 3298	. . 3 |- ( ph -> ( dom F u. dom G ) C_ ( A ... D ) )
69	52, 68	eqsstrid 3188	. 2 |- ( ph -> dom ( F u. G ) C_ ( A ... D ) )
70		isstructr 12409	. 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 1243	1 |- ( ph -> ( F u. G ) Struct <. A , D >. )

Syntax hints:   -> wi 4   /\ w3a 968   = wceq 1343   e. wcel 2136   _Vcvv 2726   \ cdif 3113   u. cun 3114   i^i cin 3115   C_ wss 3116   (/)c0 3409   {csn 3576   <.cop 3579   class class class wbr 3982   dom cdm 4604   Fun wfun 5182   ` cfv 5188  (class class class)co 5842   < clt 7933   <_ cle 7934   NNcn 8857   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944   Struct cstr 12390

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-struct 12396

This theorem is referenced by:  strle2g  12486  strle3g  12487  srngstrd  12517  lmodstrd  12528  ipsstrd  12536