Theorem strleund 12348

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 12274	. . . . 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 994	. . 3 |- ( ph -> ( A e. NN /\ B e. NN /\ A <_ B ) )
5	4	simp1d 994	. 2 |- ( ph -> A e. NN )
6		strleund.g	. . . . 5 |- ( ph -> G Struct <. C , D >. )
7		isstructim 12274	. . . . 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 994	. . 3 |- ( ph -> ( C e. NN /\ D e. NN /\ C <_ D ) )
10	9	simp2d 995	. 2 |- ( ph -> D e. NN )
11	5	nnred 8851	. . 3 |- ( ph -> A e. RR )
12	9	simp1d 994	. . . 4 |- ( ph -> C e. NN )
13	12	nnred 8851	. . 3 |- ( ph -> C e. RR )
14	10	nnred 8851	. . 3 |- ( ph -> D e. RR )
15	4	simp2d 995	. . . . 5 |- ( ph -> B e. NN )
16	15	nnred 8851	. . . 4 |- ( ph -> B e. RR )
17	4	simp3d 996	. . . 4 |- ( ph -> A <_ B )
18		strleund.l	. . . . 5 |- ( ph -> B < C )
19	16, 13, 18	ltled 7998	. . . 4 |- ( ph -> B <_ C )
20	11, 16, 13, 17, 19	letrd 8003	. . 3 |- ( ph -> A <_ C )
21	9	simp3d 996	. . 3 |- ( ph -> C <_ D )
22	11, 13, 14, 20, 21	letrd 8003	. 2 |- ( ph -> A <_ D )
23	3	simp2d 995	. . . 4 |- ( ph -> Fun ( F \ { (/) } ) )
24	8	simp2d 995	. . . 4 |- ( ph -> Fun ( G \ { (/) } ) )
25		difss 3234	. . . . . . . 8 |- ( F \ { (/) } ) C_ F
26		dmss 4787	. . . . . . . 8 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
27	25, 26	mp1i 10	. . . . . . 7 |- ( ph -> dom ( F \ { (/) } ) C_ dom F )
28	3	simp3d 996	. . . . . . 7 |- ( ph -> dom F C_ ( A ... B ) )
29	27, 28	sstrd 3138	. . . . . 6 |- ( ph -> dom ( F \ { (/) } ) C_ ( A ... B ) )
30		difss 3234	. . . . . . . 8 |- ( G \ { (/) } ) C_ G
31		dmss 4787	. . . . . . . 8 |- ( ( G \ { (/) } ) C_ G -> dom ( G \ { (/) } ) C_ dom G )
32	30, 31	mp1i 10	. . . . . . 7 |- ( ph -> dom ( G \ { (/) } ) C_ dom G )
33	8	simp3d 996	. . . . . . 7 |- ( ph -> dom G C_ ( C ... D ) )
34	32, 33	sstrd 3138	. . . . . 6 |- ( ph -> dom ( G \ { (/) } ) C_ ( C ... D ) )
35		ss2in 3336	. . . . . 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 9960	. . . . . 6 |- ( B < C -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
38	18, 37	syl 14	. . . . 5 |- ( ph -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
39		sseq0 3436	. . . . 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 5216	. . . 4 |- ( ( ( Fun ( F \ { (/) } ) /\ Fun ( G \ { (/) } ) ) /\ ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) ) -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
42	23, 24, 40, 41	syl21anc 1219	. . 3 |- ( ph -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
43		difundir 3361	. . . 4 |- ( ( F u. G ) \ { (/) } ) = ( ( F \ { (/) } ) u. ( G \ { (/) } ) )
44	43	funeqi 5193	. . 3 |- ( Fun ( ( F u. G ) \ { (/) } ) <-> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
45	42, 44	sylibr 133	. 2 |- ( ph -> Fun ( ( F u. G ) \ { (/) } ) )
46		structex 12272	. . . 4 |- ( F Struct <. A , B >. -> F e. _V )
47	1, 46	syl 14	. . 3 |- ( ph -> F e. _V )
48		structex 12272	. . . 4 |- ( G Struct <. C , D >. -> G e. _V )
49	6, 48	syl 14	. . 3 |- ( ph -> G e. _V )
50		unexg 4405	. . 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 4795	. . 3 |- dom ( F u. G ) = ( dom F u. dom G )
53	15	nnzd 9290	. . . . . . 7 |- ( ph -> B e. ZZ )
54	10	nnzd 9290	. . . . . . 7 |- ( ph -> D e. ZZ )
55	16, 13, 14, 19, 21	letrd 8003	. . . . . . 7 |- ( ph -> B <_ D )
56		eluz2 9450	. . . . . . 7 |- ( D e. ( ZZ>= ` B ) <-> ( B e. ZZ /\ D e. ZZ /\ B <_ D ) )
57	53, 54, 55, 56	syl3anbrc 1166	. . . . . 6 |- ( ph -> D e. ( ZZ>= ` B ) )
58		fzss2 9972	. . . . . 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 3138	. . . 4 |- ( ph -> dom F C_ ( A ... D ) )
61	5	nnzd 9290	. . . . . . 7 |- ( ph -> A e. ZZ )
62	12	nnzd 9290	. . . . . . 7 |- ( ph -> C e. ZZ )
63		eluz2 9450	. . . . . . 7 |- ( C e. ( ZZ>= ` A ) <-> ( A e. ZZ /\ C e. ZZ /\ A <_ C ) )
64	61, 62, 20, 63	syl3anbrc 1166	. . . . . 6 |- ( ph -> C e. ( ZZ>= ` A ) )
65		fzss1 9971	. . . . . 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 3138	. . . 4 |- ( ph -> dom G C_ ( A ... D ) )
68	60, 67	unssd 3284	. . 3 |- ( ph -> ( dom F u. dom G ) C_ ( A ... D ) )
69	52, 68	eqsstrid 3174	. 2 |- ( ph -> dom ( F u. G ) C_ ( A ... D ) )
70		isstructr 12275	. 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 1235	1 |- ( ph -> ( F u. G ) Struct <. A , D >. )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1335   e. wcel 2128   _Vcvv 2712   \ cdif 3099   u. cun 3100   i^i cin 3101   C_ wss 3102   (/)c0 3395   {csn 3561   <.cop 3564   class class class wbr 3967   dom cdm 4588   Fun wfun 5166   ` cfv 5172  (class class class)co 5826   < clt 7914   <_ cle 7915   NNcn 8838   ZZcz 9172   ZZ>=cuz 9444   ...cfz 9918   Struct cstr 12256

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-1cn 7827  ax-1re 7828  ax-icn 7829  ax-addcl 7830  ax-addrcl 7831  ax-mulcl 7832  ax-addcom 7834  ax-addass 7836  ax-distr 7838  ax-i2m1 7839  ax-0lt1 7840  ax-0id 7842  ax-rnegex 7843  ax-cnre 7845  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848  ax-pre-ltadd 7850

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920  df-sub 8052  df-neg 8053  df-inn 8839  df-n0 9096  df-z 9173  df-uz 9445  df-fz 9919  df-struct 12262

This theorem is referenced by:  strle2g  12351  strle3g  12352  srngstrd  12382  lmodstrd  12393  ipsstrd  12401