Theorem sge0xp 40646

Description: Combine two generalized sums of nonnegative extended reals into a single generalized sum over the cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
sge0xp.1	|- F/ k ph
sge0xp.z	|- ( z = <. j , k >. -> D = C )
sge0xp.a	|- ( ph -> A e. V )
sge0xp.b	|- ( ph -> B e. W )
sge0xp.d	|- ( ( ph /\ j e. A /\ k e. B ) -> C e. ( 0 [,] +oo ) )

Assertion

Ref	Expression
sge0xp	|- ( ph -> (Σ^ ` ( j e. A |-> (Σ^ ` ( k e. B |-> C ) ) ) ) = (Σ^ ` ( z e. ( A X. B ) |-> D ) ) )

Distinct variable groups:   A, j, k, z   B, j, k, z   z, C   D, j, k   ph, j, z

Allowed substitution hints:   ph( k)   C( j, k)   D( z)   V( z, j, k)   W( z, j, k)

Proof of Theorem sge0xp

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sge0xp.a	. . 3 |- ( ph -> A e. V )
2		snex 4908	. . . . . 6 |- { j } e. _V
3	2	a1i 11	. . . . 5 |- ( ph -> { j } e. _V )
4		sge0xp.b	. . . . 5 |- ( ph -> B e. W )
5		xpexg 6960	. . . . 5 |- ( ( { j } e. _V /\ B e. W ) -> ( { j } X. B ) e. _V )
6	3, 4, 5	syl2anc 693	. . . 4 |- ( ph -> ( { j } X. B ) e. _V )
7	6	adantr 481	. . 3 |- ( ( ph /\ j e. A ) -> ( { j } X. B ) e. _V )
8		disjsnxp 39239	. . . 4 |- Disj j e. A ( { j } X. B )
9	8	a1i 11	. . 3 |- ( ph -> Disj j e. A ( { j } X. B ) )
10		vex 3203	. . . . . . . 8 |- j e. _V
11		elsnxp 5677	. . . . . . . 8 |- ( j e. _V -> ( z e. ( { j } X. B ) <-> E. k e. B z = <. j , k >. ) )
12	10, 11	ax-mp 5	. . . . . . 7 |- ( z e. ( { j } X. B ) <-> E. k e. B z = <. j , k >. )
13	12	biimpi 206	. . . . . 6 |- ( z e. ( { j } X. B ) -> E. k e. B z = <. j , k >. )
14	13	adantl 482	. . . . 5 |- ( ( ( ph /\ j e. A ) /\ z e. ( { j } X. B ) ) -> E. k e. B z = <. j , k >. )
15		sge0xp.1	. . . . . . . 8 |- F/ k ph
16		nfv 1843	. . . . . . . 8 |- F/ k j e. A
17	15, 16	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ j e. A )
18		nfv 1843	. . . . . . 7 |- F/ k z e. ( { j } X. B )
19	17, 18	nfan 1828	. . . . . 6 |- F/ k ( ( ph /\ j e. A ) /\ z e. ( { j } X. B ) )
20		nfv 1843	. . . . . 6 |- F/ k D e. ( 0 [,] +oo )
21		sge0xp.z	. . . . . . . . . 10 |- ( z = <. j , k >. -> D = C )
22	21	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( ph /\ j e. A ) /\ k e. B /\ z = <. j , k >. ) -> D = C )
23		sge0xp.d	. . . . . . . . . . 11 |- ( ( ph /\ j e. A /\ k e. B ) -> C e. ( 0 [,] +oo ) )
24	23	3expa 1265	. . . . . . . . . 10 |- ( ( ( ph /\ j e. A ) /\ k e. B ) -> C e. ( 0 [,] +oo ) )
25	24	3adant3 1081	. . . . . . . . 9 |- ( ( ( ph /\ j e. A ) /\ k e. B /\ z = <. j , k >. ) -> C e. ( 0 [,] +oo ) )
26	22, 25	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ph /\ j e. A ) /\ k e. B /\ z = <. j , k >. ) -> D e. ( 0 [,] +oo ) )
27	26	3exp 1264	. . . . . . 7 |- ( ( ph /\ j e. A ) -> ( k e. B -> ( z = <. j , k >. -> D e. ( 0 [,] +oo ) ) ) )
28	27	adantr 481	. . . . . 6 |- ( ( ( ph /\ j e. A ) /\ z e. ( { j } X. B ) ) -> ( k e. B -> ( z = <. j , k >. -> D e. ( 0 [,] +oo ) ) ) )
29	19, 20, 28	rexlimd 3026	. . . . 5 |- ( ( ( ph /\ j e. A ) /\ z e. ( { j } X. B ) ) -> ( E. k e. B z = <. j , k >. -> D e. ( 0 [,] +oo ) ) )
30	14, 29	mpd 15	. . . 4 |- ( ( ( ph /\ j e. A ) /\ z e. ( { j } X. B ) ) -> D e. ( 0 [,] +oo ) )
31	30	3impa 1259	. . 3 |- ( ( ph /\ j e. A /\ z e. ( { j } X. B ) ) -> D e. ( 0 [,] +oo ) )
32	1, 7, 9, 31	sge0iunmpt 40635	. 2 |- ( ph -> (Σ^ ` ( z e. U_ j e. A ( { j } X. B ) |-> D ) ) = (Σ^ ` ( j e. A |-> (Σ^ ` ( z e. ( { j } X. B ) |-> D ) ) ) ) )
33		iunxpconst 5175	. . . . . 6 |- U_ j e. A ( { j } X. B ) = ( A X. B )
34	33	eqcomi 2631	. . . . 5 |- ( A X. B ) = U_ j e. A ( { j } X. B )
35	34	a1i 11	. . . 4 |- ( ph -> ( A X. B ) = U_ j e. A ( { j } X. B ) )
36	35	mpteq1d 4738	. . 3 |- ( ph -> ( z e. ( A X. B ) |-> D ) = ( z e. U_ j e. A ( { j } X. B ) |-> D ) )
37	36	fveq2d 6195	. 2 |- ( ph -> (Σ^ ` ( z e. ( A X. B ) |-> D ) ) = (Σ^ ` ( z e. U_ j e. A ( { j } X. B ) |-> D ) ) )
38		nfv 1843	. . . 4 |- F/ j ph
39		nfv 1843	. . . . . 6 |- F/ z ( ph /\ j e. A )
40	4	adantr 481	. . . . . 6 |- ( ( ph /\ j e. A ) -> B e. W )
41		simpr 477	. . . . . . 7 |- ( ( ph /\ j e. A ) -> j e. A )
42		eqid 2622	. . . . . . 7 |- ( i e. B |-> <. j , i >. ) = ( i e. B |-> <. j , i >. )
43	41, 42	projf1o 39386	. . . . . 6 |- ( ( ph /\ j e. A ) -> ( i e. B |-> <. j , i >. ) : B -1-1-onto-> ( { j } X. B ) )
44		eqidd 2623	. . . . . . . 8 |- ( ( ph /\ k e. B ) -> ( i e. B |-> <. j , i >. ) = ( i e. B |-> <. j , i >. ) )
45		opeq2 4403	. . . . . . . . 9 |- ( i = k -> <. j , i >. = <. j , k >. )
46	45	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ k e. B ) /\ i = k ) -> <. j , i >. = <. j , k >. )
47		simpr 477	. . . . . . . 8 |- ( ( ph /\ k e. B ) -> k e. B )
48		opex 4932	. . . . . . . . 9 |- <. j , k >. e. _V
49	48	a1i 11	. . . . . . . 8 |- ( ( ph /\ k e. B ) -> <. j , k >. e. _V )
50	44, 46, 47, 49	fvmptd 6288	. . . . . . 7 |- ( ( ph /\ k e. B ) -> ( ( i e. B |-> <. j , i >. ) ` k ) = <. j , k >. )
51	50	adantlr 751	. . . . . 6 |- ( ( ( ph /\ j e. A ) /\ k e. B ) -> ( ( i e. B |-> <. j , i >. ) ` k ) = <. j , k >. )
52	39, 17, 21, 40, 43, 51, 30	sge0f1o 40599	. . . . 5 |- ( ( ph /\ j e. A ) -> (Σ^ ` ( z e. ( { j } X. B ) |-> D ) ) = (Σ^ ` ( k e. B |-> C ) ) )
53	52	eqcomd 2628	. . . 4 |- ( ( ph /\ j e. A ) -> (Σ^ ` ( k e. B |-> C ) ) = (Σ^ ` ( z e. ( { j } X. B ) |-> D ) ) )
54	38, 53	mpteq2da 4743	. . 3 |- ( ph -> ( j e. A |-> (Σ^ ` ( k e. B |-> C ) ) ) = ( j e. A |-> (Σ^ ` ( z e. ( { j } X. B ) |-> D ) ) ) )
55	54	fveq2d 6195	. 2 |- ( ph -> (Σ^ ` ( j e. A |-> (Σ^ ` ( k e. B |-> C ) ) ) ) = (Σ^ ` ( j e. A |-> (Σ^ ` ( z e. ( { j } X. B ) |-> D ) ) ) ) )
56	32, 37, 55	3eqtr4rd 2667	1 |- ( ph -> (Σ^ ` ( j e. A |-> (Σ^ ` ( k e. B |-> C ) ) ) ) = (Σ^ ` ( z e. ( A X. B ) |-> D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177   <.cop 4183   U_ciun 4520  Disj wdisj 4620   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   [,]cicc 12178  Σ^{^}csumge0 40579

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-inf2 8538  ax-ac2 9285  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-disj 4621  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-sumge0 40580

This theorem is referenced by:  ovnsubaddlem1  40784