Theorem sge0pr 40611

Description: Sum of a pair of nonnegative extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0pr.a	|- ( ph -> A e. V )
sge0pr.b	|- ( ph -> B e. W )
sge0pr.d	|- ( ph -> D e. ( 0 [,] +oo ) )
sge0pr.e	|- ( ph -> E e. ( 0 [,] +oo ) )
sge0pr.cd	|- ( k = A -> C = D )
sge0pr.ce	|- ( k = B -> C = E )
sge0pr.ab	|- ( ph -> A =/= B )

Assertion

Ref	Expression
sge0pr	|- ( ph -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )

Distinct variable groups:   A, k   B, k   D, k   k, E   k, V   k, W   ph, k

Allowed substitution hint:   C( k)

Proof of Theorem sge0pr

Step	Hyp	Ref	Expression
1		iccssxr 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
2		sge0pr.e	. . . . . . 7 |- ( ph -> E e. ( 0 [,] +oo ) )
3	1, 2	sseldi 3601	. . . . . 6 |- ( ph -> E e. RR* )
4		mnfxr 10096	. . . . . . . 8 |- -oo e. RR*
5	4	a1i 11	. . . . . . 7 |- ( ph -> -oo e. RR* )
6		0xr 10086	. . . . . . . . 9 |- 0 e. RR*
7	6	a1i 11	. . . . . . . 8 |- ( ph -> 0 e. RR* )
8		mnflt0 11959	. . . . . . . . 9 |- -oo < 0
9	8	a1i 11	. . . . . . . 8 |- ( ph -> -oo < 0 )
10		pnfxr 10092	. . . . . . . . . 10 |- +oo e. RR*
11	10	a1i 11	. . . . . . . . 9 |- ( ph -> +oo e. RR* )
12		iccgelb 12230	. . . . . . . . 9 |- ( ( 0 e. RR* /\ +oo e. RR* /\ E e. ( 0 [,] +oo ) ) -> 0 <_ E )
13	7, 11, 2, 12	syl3anc 1326	. . . . . . . 8 |- ( ph -> 0 <_ E )
14	5, 7, 3, 9, 13	xrltletrd 11992	. . . . . . 7 |- ( ph -> -oo < E )
15	5, 3, 14	xrgtned 39538	. . . . . 6 |- ( ph -> E =/= -oo )
16		xaddpnf2 12058	. . . . . 6 |- ( ( E e. RR* /\ E =/= -oo ) -> ( +oo +e E ) = +oo )
17	3, 15, 16	syl2anc 693	. . . . 5 |- ( ph -> ( +oo +e E ) = +oo )
18	17	eqcomd 2628	. . . 4 |- ( ph -> +oo = ( +oo +e E ) )
19	18	adantr 481	. . 3 |- ( ( ph /\ D = +oo ) -> +oo = ( +oo +e E ) )
20		prex 4909	. . . . 5 |- { A , B } e. _V
21	20	a1i 11	. . . 4 |- ( ( ph /\ D = +oo ) -> { A , B } e. _V )
22		sge0pr.cd	. . . . . . . . . 10 |- ( k = A -> C = D )
23	22	adantl 482	. . . . . . . . 9 |- ( ( ph /\ k = A ) -> C = D )
24		sge0pr.d	. . . . . . . . . 10 |- ( ph -> D e. ( 0 [,] +oo ) )
25	24	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k = A ) -> D e. ( 0 [,] +oo ) )
26	23, 25	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ k = A ) -> C e. ( 0 [,] +oo ) )
27	26	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ k e. { A , B } ) /\ k = A ) -> C e. ( 0 [,] +oo ) )
28		simpll 790	. . . . . . . 8 |- ( ( ( ph /\ k e. { A , B } ) /\ -. k = A ) -> ph )
29		simpl 473	. . . . . . . . . 10 |- ( ( k e. { A , B } /\ -. k = A ) -> k e. { A , B } )
30		neqne 2802	. . . . . . . . . . 11 |- ( -. k = A -> k =/= A )
31	30	adantl 482	. . . . . . . . . 10 |- ( ( k e. { A , B } /\ -. k = A ) -> k =/= A )
32		elprn1 39865	. . . . . . . . . 10 |- ( ( k e. { A , B } /\ k =/= A ) -> k = B )
33	29, 31, 32	syl2anc 693	. . . . . . . . 9 |- ( ( k e. { A , B } /\ -. k = A ) -> k = B )
34	33	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ k e. { A , B } ) /\ -. k = A ) -> k = B )
35		sge0pr.ce	. . . . . . . . . 10 |- ( k = B -> C = E )
36	35	adantl 482	. . . . . . . . 9 |- ( ( ph /\ k = B ) -> C = E )
37	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ k = B ) -> E e. ( 0 [,] +oo ) )
38	36, 37	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ k = B ) -> C e. ( 0 [,] +oo ) )
39	28, 34, 38	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ k e. { A , B } ) /\ -. k = A ) -> C e. ( 0 [,] +oo ) )
40	27, 39	pm2.61dan 832	. . . . . 6 |- ( ( ph /\ k e. { A , B } ) -> C e. ( 0 [,] +oo ) )
41		eqid 2622	. . . . . 6 |- ( k e. { A , B } |-> C ) = ( k e. { A , B } |-> C )
42	40, 41	fmptd 6385	. . . . 5 |- ( ph -> ( k e. { A , B } |-> C ) : { A , B } --> ( 0 [,] +oo ) )
43	42	adantr 481	. . . 4 |- ( ( ph /\ D = +oo ) -> ( k e. { A , B } |-> C ) : { A , B } --> ( 0 [,] +oo ) )
44		id 22	. . . . . . 7 |- ( D = +oo -> D = +oo )
45	44	eqcomd 2628	. . . . . 6 |- ( D = +oo -> +oo = D )
46	45	adantl 482	. . . . 5 |- ( ( ph /\ D = +oo ) -> +oo = D )
47		prid1g 4295	. . . . . . . 8 |- ( D e. ( 0 [,] +oo ) -> D e. { D , E } )
48	24, 47	syl 17	. . . . . . 7 |- ( ph -> D e. { D , E } )
49		sge0pr.a	. . . . . . . . 9 |- ( ph -> A e. V )
50		sge0pr.b	. . . . . . . . 9 |- ( ph -> B e. W )
51	49, 50, 41, 22, 35	rnmptpr 39358	. . . . . . . 8 |- ( ph -> ran ( k e. { A , B } |-> C ) = { D , E } )
52	51	eqcomd 2628	. . . . . . 7 |- ( ph -> { D , E } = ran ( k e. { A , B } |-> C ) )
53	48, 52	eleqtrd 2703	. . . . . 6 |- ( ph -> D e. ran ( k e. { A , B } |-> C ) )
54	53	adantr 481	. . . . 5 |- ( ( ph /\ D = +oo ) -> D e. ran ( k e. { A , B } |-> C ) )
55	46, 54	eqeltrd 2701	. . . 4 |- ( ( ph /\ D = +oo ) -> +oo e. ran ( k e. { A , B } |-> C ) )
56	21, 43, 55	sge0pnfval 40590	. . 3 |- ( ( ph /\ D = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = +oo )
57		oveq1 6657	. . . 4 |- ( D = +oo -> ( D +e E ) = ( +oo +e E ) )
58	57	adantl 482	. . 3 |- ( ( ph /\ D = +oo ) -> ( D +e E ) = ( +oo +e E ) )
59	19, 56, 58	3eqtr4d 2666	. 2 |- ( ( ph /\ D = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )
60	1, 24	sseldi 3601	. . . . . . . 8 |- ( ph -> D e. RR* )
61		iccgelb 12230	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ +oo e. RR* /\ D e. ( 0 [,] +oo ) ) -> 0 <_ D )
62	7, 11, 24, 61	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> 0 <_ D )
63	5, 7, 60, 9, 62	xrltletrd 11992	. . . . . . . . 9 |- ( ph -> -oo < D )
64	5, 60, 63	xrgtned 39538	. . . . . . . 8 |- ( ph -> D =/= -oo )
65		xaddpnf1 12057	. . . . . . . 8 |- ( ( D e. RR* /\ D =/= -oo ) -> ( D +e +oo ) = +oo )
66	60, 64, 65	syl2anc 693	. . . . . . 7 |- ( ph -> ( D +e +oo ) = +oo )
67	66	eqcomd 2628	. . . . . 6 |- ( ph -> +oo = ( D +e +oo ) )
68	67	adantr 481	. . . . 5 |- ( ( ph /\ E = +oo ) -> +oo = ( D +e +oo ) )
69	20	a1i 11	. . . . . 6 |- ( ( ph /\ E = +oo ) -> { A , B } e. _V )
70	42	adantr 481	. . . . . 6 |- ( ( ph /\ E = +oo ) -> ( k e. { A , B } |-> C ) : { A , B } --> ( 0 [,] +oo ) )
71		id 22	. . . . . . . . 9 |- ( E = +oo -> E = +oo )
72	71	eqcomd 2628	. . . . . . . 8 |- ( E = +oo -> +oo = E )
73	72	adantl 482	. . . . . . 7 |- ( ( ph /\ E = +oo ) -> +oo = E )
74		prid2g 4296	. . . . . . . . . 10 |- ( E e. ( 0 [,] +oo ) -> E e. { D , E } )
75	2, 74	syl 17	. . . . . . . . 9 |- ( ph -> E e. { D , E } )
76	75, 52	eleqtrd 2703	. . . . . . . 8 |- ( ph -> E e. ran ( k e. { A , B } |-> C ) )
77	76	adantr 481	. . . . . . 7 |- ( ( ph /\ E = +oo ) -> E e. ran ( k e. { A , B } |-> C ) )
78	73, 77	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ E = +oo ) -> +oo e. ran ( k e. { A , B } |-> C ) )
79	69, 70, 78	sge0pnfval 40590	. . . . 5 |- ( ( ph /\ E = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = +oo )
80		oveq2 6658	. . . . . 6 |- ( E = +oo -> ( D +e E ) = ( D +e +oo ) )
81	80	adantl 482	. . . . 5 |- ( ( ph /\ E = +oo ) -> ( D +e E ) = ( D +e +oo ) )
82	68, 79, 81	3eqtr4d 2666	. . . 4 |- ( ( ph /\ E = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )
83	82	adantlr 751	. . 3 |- ( ( ( ph /\ -. D = +oo ) /\ E = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )
84		rge0ssre 12280	. . . . . . . 8 |- ( 0 [,) +oo ) C_ RR
85		ax-resscn 9993	. . . . . . . 8 |- RR C_ CC
86	84, 85	sstri 3612	. . . . . . 7 |- ( 0 [,) +oo ) C_ CC
87	6	a1i 11	. . . . . . . . 9 |- ( ( ph /\ -. D = +oo ) -> 0 e. RR* )
88	10	a1i 11	. . . . . . . . 9 |- ( ( ph /\ -. D = +oo ) -> +oo e. RR* )
89	60	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. D = +oo ) -> D e. RR* )
90	62	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. D = +oo ) -> 0 <_ D )
91		pnfge 11964	. . . . . . . . . . . 12 |- ( D e. RR* -> D <_ +oo )
92	60, 91	syl 17	. . . . . . . . . . 11 |- ( ph -> D <_ +oo )
93	92	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. D = +oo ) -> D <_ +oo )
94	44	necon3bi 2820	. . . . . . . . . . 11 |- ( -. D = +oo -> D =/= +oo )
95	94	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ -. D = +oo ) -> D =/= +oo )
96	89, 88, 93, 95	xrleneltd 39539	. . . . . . . . 9 |- ( ( ph /\ -. D = +oo ) -> D < +oo )
97	87, 88, 89, 90, 96	elicod 12224	. . . . . . . 8 |- ( ( ph /\ -. D = +oo ) -> D e. ( 0 [,) +oo ) )
98	97	adantr 481	. . . . . . 7 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> D e. ( 0 [,) +oo ) )
99	86, 98	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> D e. CC )
100	6	a1i 11	. . . . . . . . 9 |- ( ( ph /\ -. E = +oo ) -> 0 e. RR* )
101	10	a1i 11	. . . . . . . . 9 |- ( ( ph /\ -. E = +oo ) -> +oo e. RR* )
102	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. E = +oo ) -> E e. RR* )
103	13	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. E = +oo ) -> 0 <_ E )
104		pnfge 11964	. . . . . . . . . . . 12 |- ( E e. RR* -> E <_ +oo )
105	3, 104	syl 17	. . . . . . . . . . 11 |- ( ph -> E <_ +oo )
106	105	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. E = +oo ) -> E <_ +oo )
107	71	necon3bi 2820	. . . . . . . . . . 11 |- ( -. E = +oo -> E =/= +oo )
108	107	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ -. E = +oo ) -> E =/= +oo )
109	102, 101, 106, 108	xrleneltd 39539	. . . . . . . . 9 |- ( ( ph /\ -. E = +oo ) -> E < +oo )
110	100, 101, 102, 103, 109	elicod 12224	. . . . . . . 8 |- ( ( ph /\ -. E = +oo ) -> E e. ( 0 [,) +oo ) )
111	86, 110	sseldi 3601	. . . . . . 7 |- ( ( ph /\ -. E = +oo ) -> E e. CC )
112	111	adantlr 751	. . . . . 6 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> E e. CC )
113	99, 112	jca 554	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> ( D e. CC /\ E e. CC ) )
114	49, 50	jca 554	. . . . . 6 |- ( ph -> ( A e. V /\ B e. W ) )
115	114	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> ( A e. V /\ B e. W ) )
116		sge0pr.ab	. . . . . 6 |- ( ph -> A =/= B )
117	116	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> A =/= B )
118	22, 35, 113, 115, 117	sumpr 14477	. . . 4 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> sum_ k e. { A , B } C = ( D + E ) )
119		prfi 8235	. . . . . 6 |- { A , B } e. Fin
120	119	a1i 11	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> { A , B } e. Fin )
121	22	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ -. D = +oo ) /\ k = A ) -> C = D )
122	97	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ -. D = +oo ) /\ k = A ) -> D e. ( 0 [,) +oo ) )
123	121, 122	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ -. D = +oo ) /\ k = A ) -> C e. ( 0 [,) +oo ) )
124	123	ad4ant14 1293	. . . . . 6 |- ( ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) /\ k = A ) -> C e. ( 0 [,) +oo ) )
125		simp-4l 806	. . . . . . 7 |- ( ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) /\ -. k = A ) -> ph )
126		simpllr 799	. . . . . . 7 |- ( ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) /\ -. k = A ) -> -. E = +oo )
127	33	adantll 750	. . . . . . 7 |- ( ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) /\ -. k = A ) -> k = B )
128	36	3adant2 1080	. . . . . . . 8 |- ( ( ph /\ -. E = +oo /\ k = B ) -> C = E )
129	110	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ -. E = +oo /\ k = B ) -> E e. ( 0 [,) +oo ) )
130	128, 129	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ -. E = +oo /\ k = B ) -> C e. ( 0 [,) +oo ) )
131	125, 126, 127, 130	syl3anc 1326	. . . . . 6 |- ( ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) /\ -. k = A ) -> C e. ( 0 [,) +oo ) )
132	124, 131	pm2.61dan 832	. . . . 5 |- ( ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) /\ k e. { A , B } ) -> C e. ( 0 [,) +oo ) )
133	120, 132	sge0fsummpt 40607	. . . 4 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = sum_ k e. { A , B } C )
134	84, 98	sseldi 3601	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> D e. RR )
135	84, 110	sseldi 3601	. . . . . 6 |- ( ( ph /\ -. E = +oo ) -> E e. RR )
136	135	adantlr 751	. . . . 5 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> E e. RR )
137		rexadd 12063	. . . . 5 |- ( ( D e. RR /\ E e. RR ) -> ( D +e E ) = ( D + E ) )
138	134, 136, 137	syl2anc 693	. . . 4 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> ( D +e E ) = ( D + E ) )
139	118, 133, 138	3eqtr4d 2666	. . 3 |- ( ( ( ph /\ -. D = +oo ) /\ -. E = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )
140	83, 139	pm2.61dan 832	. 2 |- ( ( ph /\ -. D = +oo ) -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )
141	59, 140	pm2.61dan 832	1 |- ( ph -> (Σ^ ` ( k e. { A , B } |-> C ) ) = ( D +e E ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {cpr 4179   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   +ecxad 11944   [,)cico 12177   [,]cicc 12178   sum_csu 14416  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-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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-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:  sge0prle  40618  meadjun  40679  ovnsubadd2lem  40859