Theorem xrofsup 24079

Description: The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)

Hypotheses

Ref	Expression
xrofsup.1	|- ( ph -> X C_ RR* )
xrofsup.2	|- ( ph -> Y C_ RR* )
xrofsup.3	|- ( ph -> sup ( X , RR* , < ) =/= -oo )
xrofsup.4	|- ( ph -> sup ( Y , RR* , < ) =/= -oo )
xrofsup.5	|- ( ph -> Z = ( + e " ( X X. Y ) ) )

Assertion

Ref	Expression
xrofsup	|- ( ph -> sup ( Z , RR* , < ) = ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )

Proof of Theorem xrofsup

Dummy variables a b k u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrofsup.5	. . 3 |- ( ph -> Z = ( + e " ( X X. Y ) ) )
2		xrofsup.1	. . . . . . . . . 10 |- ( ph -> X C_ RR* )
3	2	sseld 3307	. . . . . . . . 9 |- ( ph -> ( x e. X -> x e. RR* ) )
4		xrofsup.2	. . . . . . . . . 10 |- ( ph -> Y C_ RR* )
5	4	sseld 3307	. . . . . . . . 9 |- ( ph -> ( y e. Y -> y e. RR* ) )
6	3, 5	anim12d 547	. . . . . . . 8 |- ( ph -> ( ( x e. X /\ y e. Y ) -> ( x e. RR* /\ y e. RR* ) ) )
7	6	imp 419	. . . . . . 7 |- ( ( ph /\ ( x e. X /\ y e. Y ) ) -> ( x e. RR* /\ y e. RR* ) )
8		xaddcl 10779	. . . . . . 7 |- ( ( x e. RR* /\ y e. RR* ) -> ( x + e y ) e. RR* )
9	7, 8	syl 16	. . . . . 6 |- ( ( ph /\ ( x e. X /\ y e. Y ) ) -> ( x + e y ) e. RR* )
10	9	ralrimivva 2758	. . . . 5 |- ( ph -> A. x e. X A. y e. Y ( x + e y ) e. RR* )
11		fveq2 5687	. . . . . . . 8 |- ( u = <. x , y >. -> ( + e ` u ) = ( + e ` <. x , y >. ) )
12		df-ov 6043	. . . . . . . 8 |- ( x + e y ) = ( + e ` <. x , y >. )
13	11, 12	syl6eqr 2454	. . . . . . 7 |- ( u = <. x , y >. -> ( + e ` u ) = ( x + e y ) )
14	13	eleq1d 2470	. . . . . 6 |- ( u = <. x , y >. -> ( ( + e ` u ) e. RR* <-> ( x + e y ) e. RR* ) )
15	14	ralxp 4975	. . . . 5 |- ( A. u e. ( X X. Y ) ( + e ` u ) e. RR* <-> A. x e. X A. y e. Y ( x + e y ) e. RR* )
16	10, 15	sylibr 204	. . . 4 |- ( ph -> A. u e. ( X X. Y ) ( + e ` u ) e. RR* )
17		xaddf 10766	. . . . . 6 |- + e : ( RR* X. RR* ) --> RR*
18		ffun 5552	. . . . . 6 |- ( + e : ( RR* X. RR* ) --> RR* -> Fun + e )
19	17, 18	ax-mp 8	. . . . 5 |- Fun + e
20		xpss12 4940	. . . . . . 7 |- ( ( X C_ RR* /\ Y C_ RR* ) -> ( X X. Y ) C_ ( RR* X. RR* ) )
21	2, 4, 20	syl2anc 643	. . . . . 6 |- ( ph -> ( X X. Y ) C_ ( RR* X. RR* ) )
22	17	fdmi 5555	. . . . . 6 |- dom + e = ( RR* X. RR* )
23	21, 22	syl6sseqr 3355	. . . . 5 |- ( ph -> ( X X. Y ) C_ dom + e )
24		funimass4 5736	. . . . 5 |- ( ( Fun + e /\ ( X X. Y ) C_ dom + e ) -> ( ( + e " ( X X. Y ) ) C_ RR* <-> A. u e. ( X X. Y ) ( + e ` u ) e. RR* ) )
25	19, 23, 24	sylancr 645	. . . 4 |- ( ph -> ( ( + e " ( X X. Y ) ) C_ RR* <-> A. u e. ( X X. Y ) ( + e ` u ) e. RR* ) )
26	16, 25	mpbird 224	. . 3 |- ( ph -> ( + e " ( X X. Y ) ) C_ RR* )
27	1, 26	eqsstrd 3342	. 2 |- ( ph -> Z C_ RR* )
28		supxrcl 10849	. . . 4 |- ( X C_ RR* -> sup ( X , RR* , < ) e. RR* )
29	2, 28	syl 16	. . 3 |- ( ph -> sup ( X , RR* , < ) e. RR* )
30		supxrcl 10849	. . . 4 |- ( Y C_ RR* -> sup ( Y , RR* , < ) e. RR* )
31	4, 30	syl 16	. . 3 |- ( ph -> sup ( Y , RR* , < ) e. RR* )
32	29, 31	xaddcld 10836	. 2 |- ( ph -> ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) e. RR* )
33	1	eleq2d 2471	. . . . 5 |- ( ph -> ( z e. Z <-> z e. ( + e " ( X X. Y ) ) ) )
34	33	pm5.32i 619	. . . 4 |- ( ( ph /\ z e. Z ) <-> ( ph /\ z e. ( + e " ( X X. Y ) ) ) )
35		nfvd 1627	. . . . 5 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> F/ x z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
36		nfvd 1627	. . . . 5 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> F/ y z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
37	2	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> X C_ RR* )
38		simprl 733	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> x e. X )
39		supxrub 10859	. . . . . . . . . . 11 |- ( ( X C_ RR* /\ x e. X ) -> x <_ sup ( X , RR* , < ) )
40	37, 38, 39	syl2anc 643	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> x <_ sup ( X , RR* , < ) )
41	4	ad2antrr 707	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> Y C_ RR* )
42		simprr 734	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> y e. Y )
43		supxrub 10859	. . . . . . . . . . 11 |- ( ( Y C_ RR* /\ y e. Y ) -> y <_ sup ( Y , RR* , < ) )
44	41, 42, 43	syl2anc 643	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> y <_ sup ( Y , RR* , < ) )
45	37, 38	sseldd 3309	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> x e. RR* )
46	41, 42	sseldd 3309	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> y e. RR* )
47	37, 28	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> sup ( X , RR* , < ) e. RR* )
48	41, 30	syl 16	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> sup ( Y , RR* , < ) e. RR* )
49		xle2add 10794	. . . . . . . . . . 11 |- ( ( ( x e. RR* /\ y e. RR* ) /\ ( sup ( X , RR* , < ) e. RR* /\ sup ( Y , RR* , < ) e. RR* ) ) -> ( ( x <_ sup ( X , RR* , < ) /\ y <_ sup ( Y , RR* , < ) ) -> ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
50	45, 46, 47, 48, 49	syl22anc 1185	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> ( ( x <_ sup ( X , RR* , < ) /\ y <_ sup ( Y , RR* , < ) ) -> ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
51	40, 44, 50	mp2and 661	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) /\ ( x e. X /\ y e. Y ) ) -> ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
52	51	ralrimivva 2758	. . . . . . . 8 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> A. x e. X A. y e. Y ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
53		fvelima 5737	. . . . . . . . . . 11 |- ( ( Fun + e /\ z e. ( + e " ( X X. Y ) ) ) -> E. u e. ( X X. Y ) ( + e ` u ) = z )
54	19, 53	mpan 652	. . . . . . . . . 10 |- ( z e. ( + e " ( X X. Y ) ) -> E. u e. ( X X. Y ) ( + e ` u ) = z )
55	54	adantl 453	. . . . . . . . 9 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> E. u e. ( X X. Y ) ( + e ` u ) = z )
56	13	eqeq1d 2412	. . . . . . . . . 10 |- ( u = <. x , y >. -> ( ( + e ` u ) = z <-> ( x + e y ) = z ) )
57	56	rexxp 4976	. . . . . . . . 9 |- ( E. u e. ( X X. Y ) ( + e ` u ) = z <-> E. x e. X E. y e. Y ( x + e y ) = z )
58	55, 57	sylib 189	. . . . . . . 8 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> E. x e. X E. y e. Y ( x + e y ) = z )
59	52, 58	r19.29d2r 2811	. . . . . . 7 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> E. x e. X E. y e. Y ( ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) /\ ( x + e y ) = z ) )
60		ancom 438	. . . . . . . 8 |- ( ( ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) /\ ( x + e y ) = z ) <-> ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
61	60	2rexbii 2693	. . . . . . 7 |- ( E. x e. X E. y e. Y ( ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) /\ ( x + e y ) = z ) <-> E. x e. X E. y e. Y ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
62	59, 61	sylib 189	. . . . . 6 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> E. x e. X E. y e. Y ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
63		breq1 4175	. . . . . . . . 9 |- ( ( x + e y ) = z -> ( ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) <-> z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) )
64	63	biimpa 471	. . . . . . . 8 |- ( ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
65	64	reximi 2773	. . . . . . 7 |- ( E. y e. Y ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> E. y e. Y z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
66	65	reximi 2773	. . . . . 6 |- ( E. x e. X E. y e. Y ( ( x + e y ) = z /\ ( x + e y ) <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> E. x e. X E. y e. Y z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
67	62, 66	syl 16	. . . . 5 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> E. x e. X E. y e. Y z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
68	35, 36, 67	19.9d2r 23922	. . . 4 |- ( ( ph /\ z e. ( + e " ( X X. Y ) ) ) -> z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
69	34, 68	sylbi 188	. . 3 |- ( ( ph /\ z e. Z ) -> z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
70	69	ralrimiva 2749	. 2 |- ( ph -> A. z e. Z z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
71	2	ad2antrr 707	. . . . . 6 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> X C_ RR* )
72	4	ad2antrr 707	. . . . . 6 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> Y C_ RR* )
73		simplr 732	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> z e. RR )
74	29	ad2antrr 707	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> sup ( X , RR* , < ) e. RR* )
75	31	ad2antrr 707	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> sup ( Y , RR* , < ) e. RR* )
76		xrofsup.3	. . . . . . . 8 |- ( ph -> sup ( X , RR* , < ) =/= -oo )
77	76	ad2antrr 707	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> sup ( X , RR* , < ) =/= -oo )
78		xrofsup.4	. . . . . . . 8 |- ( ph -> sup ( Y , RR* , < ) =/= -oo )
79	78	ad2antrr 707	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> sup ( Y , RR* , < ) =/= -oo )
80		simpr 448	. . . . . . 7 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
81	73, 74, 75, 77, 79, 80	xlt2addrd 24077	. . . . . 6 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) )
82		nfv 1626	. . . . . . . 8 |- F/ b ( X C_ RR* /\ Y C_ RR* )
83		nfcv 2540	. . . . . . . . 9 |- F/_ b RR*
84		nfre1 2722	. . . . . . . . 9 |- F/ b E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) )
85	83, 84	nfrex 2721	. . . . . . . 8 |- F/ b E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) )
86	82, 85	nfan 1842	. . . . . . 7 |- F/ b ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) )
87		nfvd 1627	. . . . . . 7 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> F/ a E. v e. X E. w e. Y z < ( v + e w ) )
88		nfvd 1627	. . . . . . 7 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> F/ b E. v e. X E. w e. Y z < ( v + e w ) )
89		id 20	. . . . . . . . . . . 12 |- ( ( X C_ RR* /\ Y C_ RR* ) -> ( X C_ RR* /\ Y C_ RR* ) )
90	89	ralrimivw 2750	. . . . . . . . . . 11 |- ( ( X C_ RR* /\ Y C_ RR* ) -> A. b e. RR* ( X C_ RR* /\ Y C_ RR* ) )
91	90	ralrimivw 2750	. . . . . . . . . 10 |- ( ( X C_ RR* /\ Y C_ RR* ) -> A. a e. RR* A. b e. RR* ( X C_ RR* /\ Y C_ RR* ) )
92	91	adantr 452	. . . . . . . . 9 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> A. a e. RR* A. b e. RR* ( X C_ RR* /\ Y C_ RR* ) )
93		simpr 448	. . . . . . . . 9 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) )
94	92, 93	r19.29d2r 2811	. . . . . . . 8 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. a e. RR* E. b e. RR* ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) )
95		simplll 735	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> X C_ RR* )
96		simprl 733	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> a e. RR* )
97		simplr2 1000	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> a < sup ( X , RR* , < ) )
98		supxrlub 10860	. . . . . . . . . . . . . . . 16 |- ( ( X C_ RR* /\ a e. RR* ) -> ( a < sup ( X , RR* , < ) <-> E. v e. X a < v ) )
99	98	biimpa 471	. . . . . . . . . . . . . . 15 |- ( ( ( X C_ RR* /\ a e. RR* ) /\ a < sup ( X , RR* , < ) ) -> E. v e. X a < v )
100	95, 96, 97, 99	syl21anc 1183	. . . . . . . . . . . . . 14 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> E. v e. X a < v )
101		simpllr 736	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> Y C_ RR* )
102		simprr 734	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> b e. RR* )
103		simplr3 1001	. . . . . . . . . . . . . . 15 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> b < sup ( Y , RR* , < ) )
104		supxrlub 10860	. . . . . . . . . . . . . . . 16 |- ( ( Y C_ RR* /\ b e. RR* ) -> ( b < sup ( Y , RR* , < ) <-> E. w e. Y b < w ) )
105	104	biimpa 471	. . . . . . . . . . . . . . 15 |- ( ( ( Y C_ RR* /\ b e. RR* ) /\ b < sup ( Y , RR* , < ) ) -> E. w e. Y b < w )
106	101, 102, 103, 105	syl21anc 1183	. . . . . . . . . . . . . 14 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> E. w e. Y b < w )
107		reeanv 2835	. . . . . . . . . . . . . 14 |- ( E. v e. X E. w e. Y ( a < v /\ b < w ) <-> ( E. v e. X a < v /\ E. w e. Y b < w ) )
108	100, 106, 107	sylanbrc 646	. . . . . . . . . . . . 13 |- ( ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) /\ ( a e. RR* /\ b e. RR* ) ) -> E. v e. X E. w e. Y ( a < v /\ b < w ) )
109	108	ancoms 440	. . . . . . . . . . . 12 |- ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) -> E. v e. X E. w e. Y ( a < v /\ b < w ) )
110		simplrr 738	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ ( v e. X /\ w e. Y /\ ( a < v /\ b < w ) ) ) -> ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) )
111	110	3anassrs 1175	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) )
112	111	simp1d 969	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> z = ( a + e b ) )
113		simp-4l 743	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( a e. RR* /\ b e. RR* ) )
114		simplrl 737	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ ( v e. X /\ w e. Y /\ ( a < v /\ b < w ) ) ) -> ( X C_ RR* /\ Y C_ RR* ) )
115	114	3anassrs 1175	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( X C_ RR* /\ Y C_ RR* ) )
116	115	simpld 446	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> X C_ RR* )
117		simpllr 736	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> v e. X )
118	116, 117	sseldd 3309	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> v e. RR* )
119	115	simprd 450	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> Y C_ RR* )
120		simplr 732	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> w e. Y )
121	119, 120	sseldd 3309	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> w e. RR* )
122	113, 118, 121	jca32 522	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( ( a e. RR* /\ b e. RR* ) /\ ( v e. RR* /\ w e. RR* ) ) )
123		simpr 448	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( a < v /\ b < w ) )
124	122, 123	jca 519	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> ( ( ( a e. RR* /\ b e. RR* ) /\ ( v e. RR* /\ w e. RR* ) ) /\ ( a < v /\ b < w ) ) )
125		xlt2add 10795	. . . . . . . . . . . . . . . . . 18 |- ( ( ( a e. RR* /\ b e. RR* ) /\ ( v e. RR* /\ w e. RR* ) ) -> ( ( a < v /\ b < w ) -> ( a + e b ) < ( v + e w ) ) )
126	125	imp 419	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( v e. RR* /\ w e. RR* ) ) /\ ( a < v /\ b < w ) ) -> ( a + e b ) < ( v + e w ) )
127		breq1 4175	. . . . . . . . . . . . . . . . . 18 |- ( z = ( a + e b ) -> ( z < ( v + e w ) <-> ( a + e b ) < ( v + e w ) ) )
128	127	biimpar 472	. . . . . . . . . . . . . . . . 17 |- ( ( z = ( a + e b ) /\ ( a + e b ) < ( v + e w ) ) -> z < ( v + e w ) )
129	126, 128	sylan2 461	. . . . . . . . . . . . . . . 16 |- ( ( z = ( a + e b ) /\ ( ( ( a e. RR* /\ b e. RR* ) /\ ( v e. RR* /\ w e. RR* ) ) /\ ( a < v /\ b < w ) ) ) -> z < ( v + e w ) )
130	112, 124, 129	syl2anc 643	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) /\ ( a < v /\ b < w ) ) -> z < ( v + e w ) )
131	130	ex 424	. . . . . . . . . . . . . 14 |- ( ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) /\ w e. Y ) -> ( ( a < v /\ b < w ) -> z < ( v + e w ) ) )
132	131	reximdva 2778	. . . . . . . . . . . . 13 |- ( ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) /\ v e. X ) -> ( E. w e. Y ( a < v /\ b < w ) -> E. w e. Y z < ( v + e w ) ) )
133	132	reximdva 2778	. . . . . . . . . . . 12 |- ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) -> ( E. v e. X E. w e. Y ( a < v /\ b < w ) -> E. v e. X E. w e. Y z < ( v + e w ) ) )
134	109, 133	mpd 15	. . . . . . . . . . 11 |- ( ( ( a e. RR* /\ b e. RR* ) /\ ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) ) -> E. v e. X E. w e. Y z < ( v + e w ) )
135	134	ex 424	. . . . . . . . . 10 |- ( ( a e. RR* /\ b e. RR* ) -> ( ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. v e. X E. w e. Y z < ( v + e w ) ) )
136	135	reximdva 2778	. . . . . . . . 9 |- ( a e. RR* -> ( E. b e. RR* ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. b e. RR* E. v e. X E. w e. Y z < ( v + e w ) ) )
137	136	reximia 2771	. . . . . . . 8 |- ( E. a e. RR* E. b e. RR* ( ( X C_ RR* /\ Y C_ RR* ) /\ ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. a e. RR* E. b e. RR* E. v e. X E. w e. Y z < ( v + e w ) )
138	94, 137	syl 16	. . . . . . 7 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. a e. RR* E. b e. RR* E. v e. X E. w e. Y z < ( v + e w ) )
139	86, 87, 88, 138	19.9d2rf 23921	. . . . . 6 |- ( ( ( X C_ RR* /\ Y C_ RR* ) /\ E. a e. RR* E. b e. RR* ( z = ( a + e b ) /\ a < sup ( X , RR* , < ) /\ b < sup ( Y , RR* , < ) ) ) -> E. v e. X E. w e. Y z < ( v + e w ) )
140	71, 72, 81, 139	syl21anc 1183	. . . . 5 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> E. v e. X E. w e. Y z < ( v + e w ) )
141		simprl 733	. . . . . . . . . 10 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> v e. X )
142		simprr 734	. . . . . . . . . 10 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> w e. Y )
143	23	adantr 452	. . . . . . . . . 10 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> ( X X. Y ) C_ dom + e )
144	141, 142, 19, 143	elovimad 24004	. . . . . . . . 9 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> ( v + e w ) e. ( + e " ( X X. Y ) ) )
145	1	eleq2d 2471	. . . . . . . . . 10 |- ( ph -> ( ( v + e w ) e. Z <-> ( v + e w ) e. ( + e " ( X X. Y ) ) ) )
146	145	adantr 452	. . . . . . . . 9 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> ( ( v + e w ) e. Z <-> ( v + e w ) e. ( + e " ( X X. Y ) ) ) )
147	144, 146	mpbird 224	. . . . . . . 8 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> ( v + e w ) e. Z )
148		simpr 448	. . . . . . . . 9 |- ( ( ( ph /\ ( v e. X /\ w e. Y ) ) /\ k = ( v + e w ) ) -> k = ( v + e w ) )
149	148	breq2d 4184	. . . . . . . 8 |- ( ( ( ph /\ ( v e. X /\ w e. Y ) ) /\ k = ( v + e w ) ) -> ( z < k <-> z < ( v + e w ) ) )
150	147, 149	rspcedv 3016	. . . . . . 7 |- ( ( ph /\ ( v e. X /\ w e. Y ) ) -> ( z < ( v + e w ) -> E. k e. Z z < k ) )
151	150	rexlimdvva 2797	. . . . . 6 |- ( ph -> ( E. v e. X E. w e. Y z < ( v + e w ) -> E. k e. Z z < k ) )
152	151	ad2antrr 707	. . . . 5 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> ( E. v e. X E. w e. Y z < ( v + e w ) -> E. k e. Z z < k ) )
153	140, 152	mpd 15	. . . 4 |- ( ( ( ph /\ z e. RR ) /\ z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) ) -> E. k e. Z z < k )
154	153	ex 424	. . 3 |- ( ( ph /\ z e. RR ) -> ( z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) -> E. k e. Z z < k ) )
155	154	ralrimiva 2749	. 2 |- ( ph -> A. z e. RR ( z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) -> E. k e. Z z < k ) )
156		supxr2 10848	. 2 |- ( ( ( Z C_ RR* /\ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) e. RR* ) /\ ( A. z e. Z z <_ ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) /\ A. z e. RR ( z < ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) -> E. k e. Z z < k ) ) ) -> sup ( Z , RR* , < ) = ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )
157	27, 32, 70, 155, 156	syl22anc 1185	1 |- ( ph -> sup ( Z , RR* , < ) = ( sup ( X , RR* , < ) + e sup ( Y , RR* , < ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   C_ wss 3280   <.cop 3777   class class class wbr 4172   X. cxp 4835   dom cdm 4837   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040   supcsup 7403   RRcr 8945   -oocmnf 9074   RR*cxr 9075   < clt 9076   <_ cle 9077   + ecxad 10664

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-2 10014  df-rp 10569  df-xneg 10666  df-xadd 10667