Theorem gsumzadd 18322

Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 5-Jun-2019.)

Hypotheses

Ref	Expression
gsumzadd.b	|- B = ( Base ` G )
gsumzadd.0	|- .0. = ( 0g ` G )
gsumzadd.p	|- .+ = ( +g ` G )
gsumzadd.z	|- Z = (Cntz ` G )
gsumzadd.g	|- ( ph -> G e. Mnd )
gsumzadd.a	|- ( ph -> A e. V )
gsumzadd.fn	|- ( ph -> F finSupp .0. )
gsumzadd.hn	|- ( ph -> H finSupp .0. )
gsumzadd.s	|- ( ph -> S e. (SubMnd ` G ) )
gsumzadd.c	|- ( ph -> S C_ ( Z ` S ) )
gsumzadd.f	|- ( ph -> F : A --> S )
gsumzadd.h	|- ( ph -> H : A --> S )

Assertion

Ref	Expression
gsumzadd	|- ( ph -> ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )

Proof of Theorem gsumzadd

Dummy variables k x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzadd.b	. 2 |- B = ( Base ` G )
2		gsumzadd.0	. 2 |- .0. = ( 0g ` G )
3		gsumzadd.p	. 2 |- .+ = ( +g ` G )
4		gsumzadd.z	. 2 |- Z = (Cntz ` G )
5		gsumzadd.g	. 2 |- ( ph -> G e. Mnd )
6		gsumzadd.a	. 2 |- ( ph -> A e. V )
7		gsumzadd.fn	. 2 |- ( ph -> F finSupp .0. )
8		gsumzadd.hn	. 2 |- ( ph -> H finSupp .0. )
9		eqid 2622	. 2 |- ( ( F u. H ) supp .0. ) = ( ( F u. H ) supp .0. )
10		gsumzadd.f	. . 3 |- ( ph -> F : A --> S )
11		gsumzadd.s	. . . 4 |- ( ph -> S e. (SubMnd ` G ) )
12	1	submss 17350	. . . 4 |- ( S e. (SubMnd ` G ) -> S C_ B )
13	11, 12	syl 17	. . 3 |- ( ph -> S C_ B )
14	10, 13	fssd 6057	. 2 |- ( ph -> F : A --> B )
15		gsumzadd.h	. . 3 |- ( ph -> H : A --> S )
16	15, 13	fssd 6057	. 2 |- ( ph -> H : A --> B )
17		gsumzadd.c	. . 3 |- ( ph -> S C_ ( Z ` S ) )
18		frn 6053	. . . 4 |- ( F : A --> S -> ran F C_ S )
19	10, 18	syl 17	. . 3 |- ( ph -> ran F C_ S )
20	4	cntzidss 17770	. . 3 |- ( ( S C_ ( Z ` S ) /\ ran F C_ S ) -> ran F C_ ( Z ` ran F ) )
21	17, 19, 20	syl2anc 693	. 2 |- ( ph -> ran F C_ ( Z ` ran F ) )
22		frn 6053	. . . 4 |- ( H : A --> S -> ran H C_ S )
23	15, 22	syl 17	. . 3 |- ( ph -> ran H C_ S )
24	4	cntzidss 17770	. . 3 |- ( ( S C_ ( Z ` S ) /\ ran H C_ S ) -> ran H C_ ( Z ` ran H ) )
25	17, 23, 24	syl2anc 693	. 2 |- ( ph -> ran H C_ ( Z ` ran H ) )
26	3	submcl 17353	. . . . . . 7 |- ( ( S e. (SubMnd ` G ) /\ x e. S /\ y e. S ) -> ( x .+ y ) e. S )
27	26	3expb 1266	. . . . . 6 |- ( ( S e. (SubMnd ` G ) /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
28	11, 27	sylan 488	. . . . 5 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x .+ y ) e. S )
29		inidm 3822	. . . . 5 |- ( A i^i A ) = A
30	28, 10, 15, 6, 6, 29	off 6912	. . . 4 |- ( ph -> ( F oF .+ H ) : A --> S )
31		frn 6053	. . . 4 |- ( ( F oF .+ H ) : A --> S -> ran ( F oF .+ H ) C_ S )
32	30, 31	syl 17	. . 3 |- ( ph -> ran ( F oF .+ H ) C_ S )
33	4	cntzidss 17770	. . 3 |- ( ( S C_ ( Z ` S ) /\ ran ( F oF .+ H ) C_ S ) -> ran ( F oF .+ H ) C_ ( Z ` ran ( F oF .+ H ) ) )
34	17, 32, 33	syl2anc 693	. 2 |- ( ph -> ran ( F oF .+ H ) C_ ( Z ` ran ( F oF .+ H ) ) )
35	17	adantr 481	. . . 4 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> S C_ ( Z ` S ) )
36	13	adantr 481	. . . . 5 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> S C_ B )
37	5	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> G e. Mnd )
38		vex 3203	. . . . . . . 8 |- x e. _V
39	38	a1i 11	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> x e. _V )
40	11	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> S e. (SubMnd ` G ) )
41		simpl 473	. . . . . . . 8 |- ( ( x C_ A /\ k e. ( A \ x ) ) -> x C_ A )
42		fssres 6070	. . . . . . . 8 |- ( ( H : A --> S /\ x C_ A ) -> ( H |` x ) : x --> S )
43	15, 41, 42	syl2an 494	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( H |` x ) : x --> S )
44	25	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ran H C_ ( Z ` ran H ) )
45		resss 5422	. . . . . . . . 9 |- ( H |` x ) C_ H
46		rnss 5354	. . . . . . . . 9 |- ( ( H |` x ) C_ H -> ran ( H |` x ) C_ ran H )
47	45, 46	ax-mp 5	. . . . . . . 8 |- ran ( H |` x ) C_ ran H
48	4	cntzidss 17770	. . . . . . . 8 |- ( ( ran H C_ ( Z ` ran H ) /\ ran ( H |` x ) C_ ran H ) -> ran ( H |` x ) C_ ( Z ` ran ( H |` x ) ) )
49	44, 47, 48	sylancl 694	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ran ( H |` x ) C_ ( Z ` ran ( H |` x ) ) )
50		ffun 6048	. . . . . . . . . . 11 |- ( H : A --> S -> Fun H )
51	15, 50	syl 17	. . . . . . . . . 10 |- ( ph -> Fun H )
52		funres 5929	. . . . . . . . . 10 |- ( Fun H -> Fun ( H |` x ) )
53	51, 52	syl 17	. . . . . . . . 9 |- ( ph -> Fun ( H |` x ) )
54	53	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> Fun ( H |` x ) )
55	8	fsuppimpd 8282	. . . . . . . . . 10 |- ( ph -> ( H supp .0. ) e. Fin )
56	55	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( H supp .0. ) e. Fin )
57		fex 6490	. . . . . . . . . . . 12 |- ( ( H : A --> S /\ A e. V ) -> H e. _V )
58	15, 6, 57	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> H e. _V )
59		fvex 6201	. . . . . . . . . . . 12 |- ( 0g ` G ) e. _V
60	2, 59	eqeltri 2697	. . . . . . . . . . 11 |- .0. e. _V
61		ressuppss 7314	. . . . . . . . . . 11 |- ( ( H e. _V /\ .0. e. _V ) -> ( ( H |` x ) supp .0. ) C_ ( H supp .0. ) )
62	58, 60, 61	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( ( H |` x ) supp .0. ) C_ ( H supp .0. ) )
63	62	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( ( H |` x ) supp .0. ) C_ ( H supp .0. ) )
64		ssfi 8180	. . . . . . . . 9 |- ( ( ( H supp .0. ) e. Fin /\ ( ( H |` x ) supp .0. ) C_ ( H supp .0. ) ) -> ( ( H |` x ) supp .0. ) e. Fin )
65	56, 63, 64	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( ( H |` x ) supp .0. ) e. Fin )
66		resfunexg 6479	. . . . . . . . . . 11 |- ( ( Fun H /\ x e. _V ) -> ( H |` x ) e. _V )
67	51, 38, 66	sylancl 694	. . . . . . . . . 10 |- ( ph -> ( H |` x ) e. _V )
68		isfsupp 8279	. . . . . . . . . 10 |- ( ( ( H |` x ) e. _V /\ .0. e. _V ) -> ( ( H |` x ) finSupp .0. <-> ( Fun ( H |` x ) /\ ( ( H |` x ) supp .0. ) e. Fin ) ) )
69	67, 60, 68	sylancl 694	. . . . . . . . 9 |- ( ph -> ( ( H |` x ) finSupp .0. <-> ( Fun ( H |` x ) /\ ( ( H |` x ) supp .0. ) e. Fin ) ) )
70	69	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( ( H |` x ) finSupp .0. <-> ( Fun ( H |` x ) /\ ( ( H |` x ) supp .0. ) e. Fin ) ) )
71	54, 65, 70	mpbir2and 957	. . . . . . 7 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( H |` x ) finSupp .0. )
72	2, 4, 37, 39, 40, 43, 49, 71	gsumzsubmcl 18318	. . . . . 6 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( G gsumg ( H |` x ) ) e. S )
73	72	snssd 4340	. . . . 5 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> { ( G gsumg ( H |` x ) ) } C_ S )
74	1, 4	cntz2ss 17765	. . . . 5 |- ( ( S C_ B /\ { ( G gsumg ( H |` x ) ) } C_ S ) -> ( Z ` S ) C_ ( Z ` { ( G gsumg ( H |` x ) ) } ) )
75	36, 73, 74	syl2anc 693	. . . 4 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( Z ` S ) C_ ( Z ` { ( G gsumg ( H |` x ) ) } ) )
76	35, 75	sstrd 3613	. . 3 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> S C_ ( Z ` { ( G gsumg ( H |` x ) ) } ) )
77		eldifi 3732	. . . . 5 |- ( k e. ( A \ x ) -> k e. A )
78	77	adantl 482	. . . 4 |- ( ( x C_ A /\ k e. ( A \ x ) ) -> k e. A )
79		ffvelrn 6357	. . . 4 |- ( ( F : A --> S /\ k e. A ) -> ( F ` k ) e. S )
80	10, 78, 79	syl2an 494	. . 3 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( F ` k ) e. S )
81	76, 80	sseldd 3604	. 2 |- ( ( ph /\ ( x C_ A /\ k e. ( A \ x ) ) ) -> ( F ` k ) e. ( Z ` { ( G gsumg ( H |` x ) ) } ) )
82	1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 16, 21, 25, 34, 81	gsumzaddlem 18321	1 |- ( ph -> ( G gsumg ( F oF .+ H ) ) = ( ( G gsumg F ) .+ ( G gsumg H ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   {csn 4177   class class class wbr 4653   ran crn 5115   |` cres 5116   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  SubMndcsubmnd 17334  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-cntz 17750

This theorem is referenced by:  gsumadd  18323  gsumzsplit  18327