Theorem supaddc 10990

Description: The supremum function distributes over addition in a sense similar to that in supmul1 10992. (Contributed by Brendan Leahy, 25-Sep-2017.)

Hypotheses

Ref	Expression
supadd.a1	|- ( ph -> A C_ RR )
supadd.a2	|- ( ph -> A =/= (/) )
supadd.a3	|- ( ph -> E. x e. RR A. y e. A y <_ x )
supaddc.b	|- ( ph -> B e. RR )
supaddc.c	|- C = { z | E. v e. A z = ( v + B ) }

Assertion

Ref	Expression
supaddc	|- ( ph -> ( sup ( A , RR , < ) + B ) = sup ( C , RR , < ) )

Distinct variable groups:   x, y, z, v, A   x, B, y, z, v   x, C   ph, z, v

Allowed substitution hints:   ph( x, y)   C( y, z, v)

Proof of Theorem supaddc

Dummy variables w a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . 7 |- w e. _V
2		oveq1 6657	. . . . . . . . . 10 |- ( v = a -> ( v + B ) = ( a + B ) )
3	2	eqeq2d 2632	. . . . . . . . 9 |- ( v = a -> ( z = ( v + B ) <-> z = ( a + B ) ) )
4	3	cbvrexv 3172	. . . . . . . 8 |- ( E. v e. A z = ( v + B ) <-> E. a e. A z = ( a + B ) )
5		eqeq1 2626	. . . . . . . . 9 |- ( z = w -> ( z = ( a + B ) <-> w = ( a + B ) ) )
6	5	rexbidv 3052	. . . . . . . 8 |- ( z = w -> ( E. a e. A z = ( a + B ) <-> E. a e. A w = ( a + B ) ) )
7	4, 6	syl5bb 272	. . . . . . 7 |- ( z = w -> ( E. v e. A z = ( v + B ) <-> E. a e. A w = ( a + B ) ) )
8		supaddc.c	. . . . . . 7 |- C = { z | E. v e. A z = ( v + B ) }
9	1, 7, 8	elab2 3354	. . . . . 6 |- ( w e. C <-> E. a e. A w = ( a + B ) )
10		supadd.a1	. . . . . . . . . 10 |- ( ph -> A C_ RR )
11	10	sselda 3603	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> a e. RR )
12		supadd.a2	. . . . . . . . . . 11 |- ( ph -> A =/= (/) )
13		supadd.a3	. . . . . . . . . . 11 |- ( ph -> E. x e. RR A. y e. A y <_ x )
14		suprcl 10983	. . . . . . . . . . 11 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
15	10, 12, 13, 14	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> sup ( A , RR , < ) e. RR )
16	15	adantr 481	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> sup ( A , RR , < ) e. RR )
17		supaddc.b	. . . . . . . . . 10 |- ( ph -> B e. RR )
18	17	adantr 481	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> B e. RR )
19	10, 12, 13	3jca 1242	. . . . . . . . . 10 |- ( ph -> ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) )
20		suprub 10984	. . . . . . . . . 10 |- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ a e. A ) -> a <_ sup ( A , RR , < ) )
21	19, 20	sylan 488	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> a <_ sup ( A , RR , < ) )
22	11, 16, 18, 21	leadd1dd 10641	. . . . . . . 8 |- ( ( ph /\ a e. A ) -> ( a + B ) <_ ( sup ( A , RR , < ) + B ) )
23		breq1 4656	. . . . . . . 8 |- ( w = ( a + B ) -> ( w <_ ( sup ( A , RR , < ) + B ) <-> ( a + B ) <_ ( sup ( A , RR , < ) + B ) ) )
24	22, 23	syl5ibrcom 237	. . . . . . 7 |- ( ( ph /\ a e. A ) -> ( w = ( a + B ) -> w <_ ( sup ( A , RR , < ) + B ) ) )
25	24	rexlimdva 3031	. . . . . 6 |- ( ph -> ( E. a e. A w = ( a + B ) -> w <_ ( sup ( A , RR , < ) + B ) ) )
26	9, 25	syl5bi 232	. . . . 5 |- ( ph -> ( w e. C -> w <_ ( sup ( A , RR , < ) + B ) ) )
27	26	ralrimiv 2965	. . . 4 |- ( ph -> A. w e. C w <_ ( sup ( A , RR , < ) + B ) )
28	11, 18	readdcld 10069	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> ( a + B ) e. RR )
29		eleq1a 2696	. . . . . . . . 9 |- ( ( a + B ) e. RR -> ( w = ( a + B ) -> w e. RR ) )
30	28, 29	syl 17	. . . . . . . 8 |- ( ( ph /\ a e. A ) -> ( w = ( a + B ) -> w e. RR ) )
31	30	rexlimdva 3031	. . . . . . 7 |- ( ph -> ( E. a e. A w = ( a + B ) -> w e. RR ) )
32	9, 31	syl5bi 232	. . . . . 6 |- ( ph -> ( w e. C -> w e. RR ) )
33	32	ssrdv 3609	. . . . 5 |- ( ph -> C C_ RR )
34		ovex 6678	. . . . . . . . 9 |- ( a + B ) e. _V
35	34	isseti 3209	. . . . . . . 8 |- E. w w = ( a + B )
36	35	rgenw 2924	. . . . . . 7 |- A. a e. A E. w w = ( a + B )
37		r19.2z 4060	. . . . . . 7 |- ( ( A =/= (/) /\ A. a e. A E. w w = ( a + B ) ) -> E. a e. A E. w w = ( a + B ) )
38	12, 36, 37	sylancl 694	. . . . . 6 |- ( ph -> E. a e. A E. w w = ( a + B ) )
39	9	exbii 1774	. . . . . . 7 |- ( E. w w e. C <-> E. w E. a e. A w = ( a + B ) )
40		n0 3931	. . . . . . 7 |- ( C =/= (/) <-> E. w w e. C )
41		rexcom4 3225	. . . . . . 7 |- ( E. a e. A E. w w = ( a + B ) <-> E. w E. a e. A w = ( a + B ) )
42	39, 40, 41	3bitr4i 292	. . . . . 6 |- ( C =/= (/) <-> E. a e. A E. w w = ( a + B ) )
43	38, 42	sylibr 224	. . . . 5 |- ( ph -> C =/= (/) )
44	15, 17	readdcld 10069	. . . . . 6 |- ( ph -> ( sup ( A , RR , < ) + B ) e. RR )
45		breq2 4657	. . . . . . . 8 |- ( x = ( sup ( A , RR , < ) + B ) -> ( w <_ x <-> w <_ ( sup ( A , RR , < ) + B ) ) )
46	45	ralbidv 2986	. . . . . . 7 |- ( x = ( sup ( A , RR , < ) + B ) -> ( A. w e. C w <_ x <-> A. w e. C w <_ ( sup ( A , RR , < ) + B ) ) )
47	46	rspcev 3309	. . . . . 6 |- ( ( ( sup ( A , RR , < ) + B ) e. RR /\ A. w e. C w <_ ( sup ( A , RR , < ) + B ) ) -> E. x e. RR A. w e. C w <_ x )
48	44, 27, 47	syl2anc 693	. . . . 5 |- ( ph -> E. x e. RR A. w e. C w <_ x )
49		suprleub 10989	. . . . 5 |- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ ( sup ( A , RR , < ) + B ) e. RR ) -> ( sup ( C , RR , < ) <_ ( sup ( A , RR , < ) + B ) <-> A. w e. C w <_ ( sup ( A , RR , < ) + B ) ) )
50	33, 43, 48, 44, 49	syl31anc 1329	. . . 4 |- ( ph -> ( sup ( C , RR , < ) <_ ( sup ( A , RR , < ) + B ) <-> A. w e. C w <_ ( sup ( A , RR , < ) + B ) ) )
51	27, 50	mpbird 247	. . 3 |- ( ph -> sup ( C , RR , < ) <_ ( sup ( A , RR , < ) + B ) )
52		suprcl 10983	. . . . . . . 8 |- ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) -> sup ( C , RR , < ) e. RR )
53	33, 43, 48, 52	syl3anc 1326	. . . . . . 7 |- ( ph -> sup ( C , RR , < ) e. RR )
54	53, 17, 15	ltsubaddd 10623	. . . . . 6 |- ( ph -> ( ( sup ( C , RR , < ) - B ) < sup ( A , RR , < ) <-> sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) )
55	54	biimpar 502	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) -> ( sup ( C , RR , < ) - B ) < sup ( A , RR , < ) )
56	53, 17	resubcld 10458	. . . . . . 7 |- ( ph -> ( sup ( C , RR , < ) - B ) e. RR )
57		suprlub 10987	. . . . . . 7 |- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) /\ ( sup ( C , RR , < ) - B ) e. RR ) -> ( ( sup ( C , RR , < ) - B ) < sup ( A , RR , < ) <-> E. a e. A ( sup ( C , RR , < ) - B ) < a ) )
58	10, 12, 13, 56, 57	syl31anc 1329	. . . . . 6 |- ( ph -> ( ( sup ( C , RR , < ) - B ) < sup ( A , RR , < ) <-> E. a e. A ( sup ( C , RR , < ) - B ) < a ) )
59	58	adantr 481	. . . . 5 |- ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) -> ( ( sup ( C , RR , < ) - B ) < sup ( A , RR , < ) <-> E. a e. A ( sup ( C , RR , < ) - B ) < a ) )
60	55, 59	mpbid 222	. . . 4 |- ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) -> E. a e. A ( sup ( C , RR , < ) - B ) < a )
61		rspe 3003	. . . . . . . . . . . . . 14 |- ( ( a e. A /\ w = ( a + B ) ) -> E. a e. A w = ( a + B ) )
62	61, 9	sylibr 224	. . . . . . . . . . . . 13 |- ( ( a e. A /\ w = ( a + B ) ) -> w e. C )
63	62	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ ( a e. A /\ w = ( a + B ) ) ) -> w e. C )
64		simplrr 801	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. A /\ w = ( a + B ) ) ) /\ w e. C ) -> w = ( a + B ) )
65	33, 43, 48	3jca 1242	. . . . . . . . . . . . . . 15 |- ( ph -> ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) )
66		suprub 10984	. . . . . . . . . . . . . . 15 |- ( ( ( C C_ RR /\ C =/= (/) /\ E. x e. RR A. w e. C w <_ x ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
67	65, 66	sylan 488	. . . . . . . . . . . . . 14 |- ( ( ph /\ w e. C ) -> w <_ sup ( C , RR , < ) )
68	67	adantlr 751	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( a e. A /\ w = ( a + B ) ) ) /\ w e. C ) -> w <_ sup ( C , RR , < ) )
69	64, 68	eqbrtrrd 4677	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( a e. A /\ w = ( a + B ) ) ) /\ w e. C ) -> ( a + B ) <_ sup ( C , RR , < ) )
70	63, 69	mpdan 702	. . . . . . . . . . 11 |- ( ( ph /\ ( a e. A /\ w = ( a + B ) ) ) -> ( a + B ) <_ sup ( C , RR , < ) )
71	70	expr 643	. . . . . . . . . 10 |- ( ( ph /\ a e. A ) -> ( w = ( a + B ) -> ( a + B ) <_ sup ( C , RR , < ) ) )
72	71	exlimdv 1861	. . . . . . . . 9 |- ( ( ph /\ a e. A ) -> ( E. w w = ( a + B ) -> ( a + B ) <_ sup ( C , RR , < ) ) )
73	35, 72	mpi 20	. . . . . . . 8 |- ( ( ph /\ a e. A ) -> ( a + B ) <_ sup ( C , RR , < ) )
74	73	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> ( a + B ) <_ sup ( C , RR , < ) )
75	28	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> ( a + B ) e. RR )
76	53	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> sup ( C , RR , < ) e. RR )
77	75, 76	lenltd 10183	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> ( ( a + B ) <_ sup ( C , RR , < ) <-> -. sup ( C , RR , < ) < ( a + B ) ) )
78	74, 77	mpbid 222	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> -. sup ( C , RR , < ) < ( a + B ) )
79	17	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> B e. RR )
80	11	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> a e. RR )
81	76, 79, 80	ltsubaddd 10623	. . . . . 6 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> ( ( sup ( C , RR , < ) - B ) < a <-> sup ( C , RR , < ) < ( a + B ) ) )
82	78, 81	mtbird 315	. . . . 5 |- ( ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) /\ a e. A ) -> -. ( sup ( C , RR , < ) - B ) < a )
83	82	nrexdv 3001	. . . 4 |- ( ( ph /\ sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) -> -. E. a e. A ( sup ( C , RR , < ) - B ) < a )
84	60, 83	pm2.65da 600	. . 3 |- ( ph -> -. sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) )
85	53, 44	eqleltd 10181	. . 3 |- ( ph -> ( sup ( C , RR , < ) = ( sup ( A , RR , < ) + B ) <-> ( sup ( C , RR , < ) <_ ( sup ( A , RR , < ) + B ) /\ -. sup ( C , RR , < ) < ( sup ( A , RR , < ) + B ) ) ) )
86	51, 84, 85	mpbir2and 957	. 2 |- ( ph -> sup ( C , RR , < ) = ( sup ( A , RR , < ) + B ) )
87	86	eqcomd 2628	1 |- ( ph -> ( sup ( A , RR , < ) + B ) = sup ( C , RR , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269

This theorem is referenced by:  supadd  10991  supsubc  39569