Theorem strleund 12935

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 12846	. . . . 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		strleund.g	. . . . 5 |- ( ph -> G Struct <. C , D >. )
7		isstructim 12846	. . . . 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 1012	. . 3 |- ( ph -> ( C e. NN /\ D e. NN /\ C <_ D ) )
10	9	simp2d 1013	. 2 |- ( ph -> D e. NN )
11	5	nnred 9049	. . 3 |- ( ph -> A e. RR )
12	9	simp1d 1012	. . . 4 |- ( ph -> C e. NN )
13	12	nnred 9049	. . 3 |- ( ph -> C e. RR )
14	10	nnred 9049	. . 3 |- ( ph -> D e. RR )
15	4	simp2d 1013	. . . . 5 |- ( ph -> B e. NN )
16	15	nnred 9049	. . . 4 |- ( ph -> B e. RR )
17	4	simp3d 1014	. . . 4 |- ( ph -> A <_ B )
18		strleund.l	. . . . 5 |- ( ph -> B < C )
19	16, 13, 18	ltled 8191	. . . 4 |- ( ph -> B <_ C )
20	11, 16, 13, 17, 19	letrd 8196	. . 3 |- ( ph -> A <_ C )
21	9	simp3d 1014	. . 3 |- ( ph -> C <_ D )
22	11, 13, 14, 20, 21	letrd 8196	. 2 |- ( ph -> A <_ D )
23	3	simp2d 1013	. . . 4 |- ( ph -> Fun ( F \ { (/) } ) )
24	8	simp2d 1013	. . . 4 |- ( ph -> Fun ( G \ { (/) } ) )
25		difss 3299	. . . . . . . 8 |- ( F \ { (/) } ) C_ F
26		dmss 4877	. . . . . . . 8 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
27	25, 26	mp1i 10	. . . . . . 7 |- ( ph -> dom ( F \ { (/) } ) C_ dom F )
28	3	simp3d 1014	. . . . . . 7 |- ( ph -> dom F C_ ( A ... B ) )
29	27, 28	sstrd 3203	. . . . . 6 |- ( ph -> dom ( F \ { (/) } ) C_ ( A ... B ) )
30		difss 3299	. . . . . . . 8 |- ( G \ { (/) } ) C_ G
31		dmss 4877	. . . . . . . 8 |- ( ( G \ { (/) } ) C_ G -> dom ( G \ { (/) } ) C_ dom G )
32	30, 31	mp1i 10	. . . . . . 7 |- ( ph -> dom ( G \ { (/) } ) C_ dom G )
33	8	simp3d 1014	. . . . . . 7 |- ( ph -> dom G C_ ( C ... D ) )
34	32, 33	sstrd 3203	. . . . . 6 |- ( ph -> dom ( G \ { (/) } ) C_ ( C ... D ) )
35		ss2in 3401	. . . . . 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 10174	. . . . . 6 |- ( B < C -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
38	18, 37	syl 14	. . . . 5 |- ( ph -> ( ( A ... B ) i^i ( C ... D ) ) = (/) )
39		sseq0 3502	. . . . 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 5315	. . . 4 |- ( ( ( Fun ( F \ { (/) } ) /\ Fun ( G \ { (/) } ) ) /\ ( dom ( F \ { (/) } ) i^i dom ( G \ { (/) } ) ) = (/) ) -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
42	23, 24, 40, 41	syl21anc 1249	. . 3 |- ( ph -> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
43		difundir 3426	. . . 4 |- ( ( F u. G ) \ { (/) } ) = ( ( F \ { (/) } ) u. ( G \ { (/) } ) )
44	43	funeqi 5292	. . 3 |- ( Fun ( ( F u. G ) \ { (/) } ) <-> Fun ( ( F \ { (/) } ) u. ( G \ { (/) } ) ) )
45	42, 44	sylibr 134	. 2 |- ( ph -> Fun ( ( F u. G ) \ { (/) } ) )
46		structex 12844	. . . 4 |- ( F Struct <. A , B >. -> F e. _V )
47	1, 46	syl 14	. . 3 |- ( ph -> F e. _V )
48		structex 12844	. . . 4 |- ( G Struct <. C , D >. -> G e. _V )
49	6, 48	syl 14	. . 3 |- ( ph -> G e. _V )
50		unexg 4490	. . 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 4885	. . 3 |- dom ( F u. G ) = ( dom F u. dom G )
53	15	nnzd 9494	. . . . . . 7 |- ( ph -> B e. ZZ )
54	10	nnzd 9494	. . . . . . 7 |- ( ph -> D e. ZZ )
55	16, 13, 14, 19, 21	letrd 8196	. . . . . . 7 |- ( ph -> B <_ D )
56		eluz2 9654	. . . . . . 7 |- ( D e. ( ZZ>= ` B ) <-> ( B e. ZZ /\ D e. ZZ /\ B <_ D ) )
57	53, 54, 55, 56	syl3anbrc 1184	. . . . . 6 |- ( ph -> D e. ( ZZ>= ` B ) )
58		fzss2 10186	. . . . . 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 3203	. . . 4 |- ( ph -> dom F C_ ( A ... D ) )
61	5	nnzd 9494	. . . . . . 7 |- ( ph -> A e. ZZ )
62	12	nnzd 9494	. . . . . . 7 |- ( ph -> C e. ZZ )
63		eluz2 9654	. . . . . . 7 |- ( C e. ( ZZ>= ` A ) <-> ( A e. ZZ /\ C e. ZZ /\ A <_ C ) )
64	61, 62, 20, 63	syl3anbrc 1184	. . . . . 6 |- ( ph -> C e. ( ZZ>= ` A ) )
65		fzss1 10185	. . . . . 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 3203	. . . 4 |- ( ph -> dom G C_ ( A ... D ) )
68	60, 67	unssd 3349	. . 3 |- ( ph -> ( dom F u. dom G ) C_ ( A ... D ) )
69	52, 68	eqsstrid 3239	. 2 |- ( ph -> dom ( F u. G ) C_ ( A ... D ) )
70		isstructr 12847	. 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 1265	1 |- ( ph -> ( F u. G ) Struct <. A , D >. )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2176   _Vcvv 2772   \ cdif 3163   u. cun 3164   i^i cin 3165   C_ wss 3166   (/)c0 3460   {csn 3633   <.cop 3636   class class class wbr 4044   dom cdm 4675   Fun wfun 5265   ` cfv 5271  (class class class)co 5944   < clt 8107   <_ cle 8108   NNcn 9036   ZZcz 9372   ZZ>=cuz 9648   ...cfz 10130   Struct cstr 12828

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 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

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-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-fz 10131  df-struct 12834

This theorem is referenced by:  strle2g  12939  strle3g  12940  srngstrd  12978  lmodstrd  12996  ipsstrd  13008  imasvalstrd  13102  prdsvalstrd  13103  psrvalstrd  14430