Theorem strleund 13316

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 13226	. . . . 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		strleund.g	. . . . 5 |- ( ph -> G Struct <. C , D >. )
7		isstructim 13226	. . . . 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 1036	. . 3 |- ( ph -> ( C e. NN /\ D e. NN /\ C <_ D ) )
10	9	simp2d 1037	. 2 |- ( ph -> D e. NN )
11	5	nnred 9250	. . 3 |- ( ph -> A e. RR )
12	9	simp1d 1036	. . . 4 |- ( ph -> C e. NN )
13	12	nnred 9250	. . 3 |- ( ph -> C e. RR )
14	10	nnred 9250	. . 3 |- ( ph -> D e. RR )
15	4	simp2d 1037	. . . . 5 |- ( ph -> B e. NN )
16	15	nnred 9250	. . . 4 |- ( ph -> B e. RR )
17	4	simp3d 1038	. . . 4 |- ( ph -> A <_ B )
18		strleund.l	. . . . 5 |- ( ph -> B < C )
19	16, 13, 18	ltled 8392	. . . 4 |- ( ph -> B <_ C )
20	11, 16, 13, 17, 19	letrd 8397	. . 3 |- ( ph -> A <_ C )
21	9	simp3d 1038	. . 3 |- ( ph -> C <_ D )
22	11, 13, 14, 20, 21	letrd 8397	. 2 |- ( ph -> A <_ D )
23	3	simp2d 1037	. . . 4 |- ( ph -> Fun ( F \ { (/) } ) )
24	8	simp2d 1037	. . . 4 |- ( ph -> Fun ( G \ { (/) } ) )
25		difss 3345	. . . . . . . 8 |- ( F \ { (/) } ) C_ F
26		dmss 4955	. . . . . . . 8 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
27	25, 26	mp1i 10	. . . . . . 7 |- ( ph -> dom ( F \ { (/) } ) C_ dom F )
28	3	simp3d 1038	. . . . . . 7 |- ( ph -> dom F C_ ( A ... B ) )
29	27, 28	sstrd 3248	. . . . . 6 |- ( ph -> dom ( F \ { (/) } ) C_ ( A ... B ) )
30		difss 3345	. . . . . . . 8 |- ( G \ { (/) } ) C_ G
31		dmss 4955	. . . . . . . 8 |- ( ( G \ { (/) } ) C_ G -> dom ( G \ { (/) } ) C_ dom G )
32	30, 31	mp1i 10	. . . . . . 7 |- ( ph -> dom ( G \ { (/) } ) C_ dom G )
33	8	simp3d 1038	. . . . . . 7 |- ( ph -> dom G C_ ( C ... D ) )
34	32, 33	sstrd 3248	. . . . . 6 |- ( ph -> dom ( G \ { (/) } ) C_ ( C ... D ) )
35		ss2in 3449	. . . . . 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 10386	. . . . . 6 |- ( B < C -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
38	18, 37	syl 14	. . . . 5 |- ( ph -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
39		sseq0 3550	. . . . 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 5397	. . . 4 |- ( ( ( Fun ( F \ { (/) } ) /\ Fun ( G \ { (/) } ) ) /\ ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) ) -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
42	23, 24, 40, 41	syl21anc 1273	. . 3 |- ( ph -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
43		difundir 3474	. . . 4 |- ( ( F u. G ) \ { (/) } ) = ( ( F \ { (/) } ) u. ( G \ { (/) } ) )
44	43	funeqi 5373	. . 3 |- ( Fun ( ( F u. G ) \ { (/) } ) <-> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
45	42, 44	sylibr 134	. 2 |- ( ph -> Fun ( ( F u. G ) \ { (/) } ) )
46		structex 13224	. . . 4 |- ( F Struct <. A , B >. -> F e. _V )
47	1, 46	syl 14	. . 3 |- ( ph -> F e. _V )
48		structex 13224	. . . 4 |- ( G Struct <. C , D >. -> G e. _V )
49	6, 48	syl 14	. . 3 |- ( ph -> G e. _V )
50		unexg 4564	. . 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 4963	. . 3 |- dom ( F u. G ) = ( dom F u. dom G )
53	15	nnzd 9699	. . . . . . 7 |- ( ph -> B e. ZZ )
54	10	nnzd 9699	. . . . . . 7 |- ( ph -> D e. ZZ )
55	16, 13, 14, 19, 21	letrd 8397	. . . . . . 7 |- ( ph -> B <_ D )
56		eluz2 9859	. . . . . . 7 |- ( D e. ( ZZ>= ` B ) <-> ( B e. ZZ /\ D e. ZZ /\ B <_ D ) )
57	53, 54, 55, 56	syl3anbrc 1208	. . . . . 6 |- ( ph -> D e. ( ZZ>= ` B ) )
58		fzss2 10398	. . . . . 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 3248	. . . 4 |- ( ph -> dom F C_ ( A ... D ) )
61	5	nnzd 9699	. . . . . . 7 |- ( ph -> A e. ZZ )
62	12	nnzd 9699	. . . . . . 7 |- ( ph -> C e. ZZ )
63		eluz2 9859	. . . . . . 7 |- ( C e. ( ZZ>= ` A ) <-> ( A e. ZZ /\ C e. ZZ /\ A <_ C ) )
64	61, 62, 20, 63	syl3anbrc 1208	. . . . . 6 |- ( ph -> C e. ( ZZ>= ` A ) )
65		fzss1 10397	. . . . . 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 3248	. . . 4 |- ( ph -> dom G C_ ( A ... D ) )
68	60, 67	unssd 3395	. . 3 |- ( ph -> ( dom F u. dom G ) C_ ( A ... D ) )
69	52, 68	eqsstrid 3284	. 2 |- ( ph -> dom ( F u. G ) C_ ( A ... D ) )
70		isstructr 13227	. 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 1289	1 |- ( ph -> ( F u. G ) Struct <. A , D >. )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   _Vcvv 2813   \ cdif 3208   u. cun 3209   i^i cin 3210   C_ wss 3211   (/)c0 3508   {csn 3689   <.cop 3692   class class class wbr 4109   dom cdm 4749   Fun wfun 5346   ` cfv 5352  (class class class)co 6050   < clt 8308   <_ cle 8309   NNcn 9237   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   Struct cstr 13208

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-struct 13214

This theorem is referenced by:  strle2g  13320  strle3g  13321  srngstrd  13359  lmodstrd  13377  ipsstrd  13389  imasvalstrd  13483  prdsvalstrd  13484  psrvalstrd  14816