Theorem xlt2addrd 29523

Description: If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)

Hypotheses

Ref	Expression
xlt2addrd.1	|- ( ph -> A e. RR )
xlt2addrd.2	|- ( ph -> B e. RR* )
xlt2addrd.3	|- ( ph -> C e. RR* )
xlt2addrd.4	|- ( ph -> B =/= -oo )
xlt2addrd.5	|- ( ph -> C =/= -oo )
xlt2addrd.6	|- ( ph -> A < ( B +e C ) )

Assertion

Ref	Expression
xlt2addrd	|- ( ph -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )

Distinct variable groups:   b, c, A   B, b, c   C, b, c

Allowed substitution hints:   ph( b, c)

Proof of Theorem xlt2addrd

Step	Hyp	Ref	Expression
1		xlt2addrd.1	. . . . . 6 |- ( ph -> A e. RR )
2	1	rexrd 10089	. . . . 5 |- ( ph -> A e. RR* )
3	2	ad2antrr 762	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> A e. RR* )
4		0xr 10086	. . . . 5 |- 0 e. RR*
5	4	a1i 11	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> 0 e. RR* )
6		xaddid1 12072	. . . . . 6 |- ( A e. RR* -> ( A +e 0 ) = A )
7	6	eqcomd 2628	. . . . 5 |- ( A e. RR* -> A = ( A +e 0 ) )
8	3, 7	syl 17	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> A = ( A +e 0 ) )
9	1	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> A e. RR )
10		ltpnf 11954	. . . . . 6 |- ( A e. RR -> A < +oo )
11	9, 10	syl 17	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> A < +oo )
12		simplr 792	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> B = +oo )
13	11, 12	breqtrrd 4681	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> A < B )
14		0ltpnf 11956	. . . . 5 |- 0 < +oo
15		simpr 477	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> C = +oo )
16	14, 15	syl5breqr 4691	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> 0 < C )
17		oveq1 6657	. . . . . . 7 |- ( b = A -> ( b +e c ) = ( A +e c ) )
18	17	eqeq2d 2632	. . . . . 6 |- ( b = A -> ( A = ( b +e c ) <-> A = ( A +e c ) ) )
19		breq1 4656	. . . . . 6 |- ( b = A -> ( b < B <-> A < B ) )
20	18, 19	3anbi12d 1400	. . . . 5 |- ( b = A -> ( ( A = ( b +e c ) /\ b < B /\ c < C ) <-> ( A = ( A +e c ) /\ A < B /\ c < C ) ) )
21		oveq2 6658	. . . . . . 7 |- ( c = 0 -> ( A +e c ) = ( A +e 0 ) )
22	21	eqeq2d 2632	. . . . . 6 |- ( c = 0 -> ( A = ( A +e c ) <-> A = ( A +e 0 ) ) )
23		breq1 4656	. . . . . 6 |- ( c = 0 -> ( c < C <-> 0 < C ) )
24	22, 23	3anbi13d 1401	. . . . 5 |- ( c = 0 -> ( ( A = ( A +e c ) /\ A < B /\ c < C ) <-> ( A = ( A +e 0 ) /\ A < B /\ 0 < C ) ) )
25	20, 24	rspc2ev 3324	. . . 4 |- ( ( A e. RR* /\ 0 e. RR* /\ ( A = ( A +e 0 ) /\ A < B /\ 0 < C ) ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
26	3, 5, 8, 13, 16, 25	syl113anc 1338	. . 3 |- ( ( ( ph /\ B = +oo ) /\ C = +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
27	2	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> A e. RR* )
28		xlt2addrd.3	. . . . . . . 8 |- ( ph -> C e. RR* )
29	28	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> C e. RR* )
30		1re 10039	. . . . . . . . . 10 |- 1 e. RR
31	30	rexri 10097	. . . . . . . . 9 |- 1 e. RR*
32	31	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> 1 e. RR* )
33	32	xnegcld 12130	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -e 1 e. RR* )
34	29, 33	xaddcld 12131	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C +e -e 1 ) e. RR* )
35	34	xnegcld 12130	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -e ( C +e -e 1 ) e. RR* )
36	27, 35	xaddcld 12131	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e -e ( C +e -e 1 ) ) e. RR* )
37	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> A e. RR )
38	37	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> A =/= -oo )
39		xrnepnf 11952	. . . . . . . . . . . . . . 15 |- ( ( C e. RR* /\ C =/= +oo ) <-> ( C e. RR \/ C = -oo ) )
40	39	biimpi 206	. . . . . . . . . . . . . 14 |- ( ( C e. RR* /\ C =/= +oo ) -> ( C e. RR \/ C = -oo ) )
41	29, 40	sylancom 701	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C e. RR \/ C = -oo ) )
42	41	orcomd 403	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C = -oo \/ C e. RR ) )
43		xlt2addrd.5	. . . . . . . . . . . . . 14 |- ( ph -> C =/= -oo )
44	43	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> C =/= -oo )
45	44	neneqd 2799	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -. C = -oo )
46		pm2.53 388	. . . . . . . . . . . 12 |- ( ( C = -oo \/ C e. RR ) -> ( -. C = -oo -> C e. RR ) )
47	42, 45, 46	sylc 65	. . . . . . . . . . 11 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> C e. RR )
48		rexsub 12064	. . . . . . . . . . 11 |- ( ( C e. RR /\ 1 e. RR ) -> ( C +e -e 1 ) = ( C - 1 ) )
49	47, 30, 48	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C +e -e 1 ) = ( C - 1 ) )
50		resubcl 10345	. . . . . . . . . . 11 |- ( ( C e. RR /\ 1 e. RR ) -> ( C - 1 ) e. RR )
51	47, 30, 50	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C - 1 ) e. RR )
52	49, 51	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C +e -e 1 ) e. RR )
53		rexneg 12042	. . . . . . . . 9 |- ( ( C +e -e 1 ) e. RR -> -e ( C +e -e 1 ) = -u ( C +e -e 1 ) )
54	52, 53	syl 17	. . . . . . . 8 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -e ( C +e -e 1 ) = -u ( C +e -e 1 ) )
55	52	renegcld 10457	. . . . . . . 8 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -u ( C +e -e 1 ) e. RR )
56	54, 55	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -e ( C +e -e 1 ) e. RR )
57	56	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> -e ( C +e -e 1 ) =/= -oo )
58	52	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C +e -e 1 ) =/= -oo )
59		xaddass 12079	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( -e ( C +e -e 1 ) e. RR* /\ -e ( C +e -e 1 ) =/= -oo ) /\ ( ( C +e -e 1 ) e. RR* /\ ( C +e -e 1 ) =/= -oo ) ) -> ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) = ( A +e ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) ) )
60	27, 38, 35, 57, 34, 58, 59	syl222anc 1342	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) = ( A +e ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) ) )
61		xaddcom 12071	. . . . . . . 8 |- ( ( -e ( C +e -e 1 ) e. RR* /\ ( C +e -e 1 ) e. RR* ) -> ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) = ( ( C +e -e 1 ) +e -e ( C +e -e 1 ) ) )
62	35, 34, 61	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) = ( ( C +e -e 1 ) +e -e ( C +e -e 1 ) ) )
63		xnegid 12069	. . . . . . . 8 |- ( ( C +e -e 1 ) e. RR* -> ( ( C +e -e 1 ) +e -e ( C +e -e 1 ) ) = 0 )
64	34, 63	syl 17	. . . . . . 7 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( ( C +e -e 1 ) +e -e ( C +e -e 1 ) ) = 0 )
65	62, 64	eqtrd 2656	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) = 0 )
66	65	oveq2d 6666	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e ( -e ( C +e -e 1 ) +e ( C +e -e 1 ) ) ) = ( A +e 0 ) )
67	27, 6	syl 17	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e 0 ) = A )
68	60, 66, 67	3eqtrrd 2661	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> A = ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) )
69	37, 51	resubcld 10458	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A - ( C - 1 ) ) e. RR )
70		ltpnf 11954	. . . . . 6 |- ( ( A - ( C - 1 ) ) e. RR -> ( A - ( C - 1 ) ) < +oo )
71	69, 70	syl 17	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A - ( C - 1 ) ) < +oo )
72		rexsub 12064	. . . . . . 7 |- ( ( A e. RR /\ ( C +e -e 1 ) e. RR ) -> ( A +e -e ( C +e -e 1 ) ) = ( A - ( C +e -e 1 ) ) )
73	37, 52, 72	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e -e ( C +e -e 1 ) ) = ( A - ( C +e -e 1 ) ) )
74	49	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A - ( C +e -e 1 ) ) = ( A - ( C - 1 ) ) )
75	73, 74	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e -e ( C +e -e 1 ) ) = ( A - ( C - 1 ) ) )
76		simplr 792	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> B = +oo )
77	71, 75, 76	3brtr4d 4685	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( A +e -e ( C +e -e 1 ) ) < B )
78	47	ltm1d 10956	. . . . 5 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C - 1 ) < C )
79	49, 78	eqbrtrd 4675	. . . 4 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> ( C +e -e 1 ) < C )
80		oveq1 6657	. . . . . . 7 |- ( b = ( A +e -e ( C +e -e 1 ) ) -> ( b +e c ) = ( ( A +e -e ( C +e -e 1 ) ) +e c ) )
81	80	eqeq2d 2632	. . . . . 6 |- ( b = ( A +e -e ( C +e -e 1 ) ) -> ( A = ( b +e c ) <-> A = ( ( A +e -e ( C +e -e 1 ) ) +e c ) ) )
82		breq1 4656	. . . . . 6 |- ( b = ( A +e -e ( C +e -e 1 ) ) -> ( b < B <-> ( A +e -e ( C +e -e 1 ) ) < B ) )
83	81, 82	3anbi12d 1400	. . . . 5 |- ( b = ( A +e -e ( C +e -e 1 ) ) -> ( ( A = ( b +e c ) /\ b < B /\ c < C ) <-> ( A = ( ( A +e -e ( C +e -e 1 ) ) +e c ) /\ ( A +e -e ( C +e -e 1 ) ) < B /\ c < C ) ) )
84		oveq2 6658	. . . . . . 7 |- ( c = ( C +e -e 1 ) -> ( ( A +e -e ( C +e -e 1 ) ) +e c ) = ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) )
85	84	eqeq2d 2632	. . . . . 6 |- ( c = ( C +e -e 1 ) -> ( A = ( ( A +e -e ( C +e -e 1 ) ) +e c ) <-> A = ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) ) )
86		breq1 4656	. . . . . 6 |- ( c = ( C +e -e 1 ) -> ( c < C <-> ( C +e -e 1 ) < C ) )
87	85, 86	3anbi13d 1401	. . . . 5 |- ( c = ( C +e -e 1 ) -> ( ( A = ( ( A +e -e ( C +e -e 1 ) ) +e c ) /\ ( A +e -e ( C +e -e 1 ) ) < B /\ c < C ) <-> ( A = ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) /\ ( A +e -e ( C +e -e 1 ) ) < B /\ ( C +e -e 1 ) < C ) ) )
88	83, 87	rspc2ev 3324	. . . 4 |- ( ( ( A +e -e ( C +e -e 1 ) ) e. RR* /\ ( C +e -e 1 ) e. RR* /\ ( A = ( ( A +e -e ( C +e -e 1 ) ) +e ( C +e -e 1 ) ) /\ ( A +e -e ( C +e -e 1 ) ) < B /\ ( C +e -e 1 ) < C ) ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
89	36, 34, 68, 77, 79, 88	syl113anc 1338	. . 3 |- ( ( ( ph /\ B = +oo ) /\ C =/= +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
90	26, 89	pm2.61dane 2881	. 2 |- ( ( ph /\ B = +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
91		xlt2addrd.2	. . . . . 6 |- ( ph -> B e. RR* )
92	91	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> B e. RR* )
93	31	a1i 11	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> 1 e. RR* )
94	93	xnegcld 12130	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -e 1 e. RR* )
95	92, 94	xaddcld 12131	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B +e -e 1 ) e. RR* )
96	2	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> A e. RR* )
97	95	xnegcld 12130	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -e ( B +e -e 1 ) e. RR* )
98	96, 97	xaddcld 12131	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e -e ( B +e -e 1 ) ) e. RR* )
99		xaddcom 12071	. . . . . 6 |- ( ( ( B +e -e 1 ) e. RR* /\ ( A +e -e ( B +e -e 1 ) ) e. RR* ) -> ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) = ( ( A +e -e ( B +e -e 1 ) ) +e ( B +e -e 1 ) ) )
100	95, 98, 99	syl2anc 693	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) = ( ( A +e -e ( B +e -e 1 ) ) +e ( B +e -e 1 ) ) )
101	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> A e. RR )
102	101	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> A =/= -oo )
103		simplr 792	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> B =/= +oo )
104		xrnepnf 11952	. . . . . . . . . . . . . . 15 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
105	104	biimpi 206	. . . . . . . . . . . . . 14 |- ( ( B e. RR* /\ B =/= +oo ) -> ( B e. RR \/ B = -oo ) )
106	92, 103, 105	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B e. RR \/ B = -oo ) )
107	106	orcomd 403	. . . . . . . . . . . 12 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B = -oo \/ B e. RR ) )
108		xlt2addrd.4	. . . . . . . . . . . . . 14 |- ( ph -> B =/= -oo )
109	108	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> B =/= -oo )
110	109	neneqd 2799	. . . . . . . . . . . 12 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -. B = -oo )
111		pm2.53 388	. . . . . . . . . . . 12 |- ( ( B = -oo \/ B e. RR ) -> ( -. B = -oo -> B e. RR ) )
112	107, 110, 111	sylc 65	. . . . . . . . . . 11 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> B e. RR )
113		rexsub 12064	. . . . . . . . . . 11 |- ( ( B e. RR /\ 1 e. RR ) -> ( B +e -e 1 ) = ( B - 1 ) )
114	112, 30, 113	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B +e -e 1 ) = ( B - 1 ) )
115		resubcl 10345	. . . . . . . . . . 11 |- ( ( B e. RR /\ 1 e. RR ) -> ( B - 1 ) e. RR )
116	112, 30, 115	sylancl 694	. . . . . . . . . 10 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B - 1 ) e. RR )
117	114, 116	eqeltrd 2701	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B +e -e 1 ) e. RR )
118		rexneg 12042	. . . . . . . . 9 |- ( ( B +e -e 1 ) e. RR -> -e ( B +e -e 1 ) = -u ( B +e -e 1 ) )
119	117, 118	syl 17	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -e ( B +e -e 1 ) = -u ( B +e -e 1 ) )
120	117	renegcld 10457	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -u ( B +e -e 1 ) e. RR )
121	119, 120	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -e ( B +e -e 1 ) e. RR )
122	121	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> -e ( B +e -e 1 ) =/= -oo )
123	117	renemnfd 10091	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B +e -e 1 ) =/= -oo )
124		xaddass 12079	. . . . . 6 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( -e ( B +e -e 1 ) e. RR* /\ -e ( B +e -e 1 ) =/= -oo ) /\ ( ( B +e -e 1 ) e. RR* /\ ( B +e -e 1 ) =/= -oo ) ) -> ( ( A +e -e ( B +e -e 1 ) ) +e ( B +e -e 1 ) ) = ( A +e ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) ) )
125	96, 102, 97, 122, 95, 123, 124	syl222anc 1342	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( ( A +e -e ( B +e -e 1 ) ) +e ( B +e -e 1 ) ) = ( A +e ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) ) )
126		xaddcom 12071	. . . . . . . . 9 |- ( ( -e ( B +e -e 1 ) e. RR* /\ ( B +e -e 1 ) e. RR* ) -> ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) = ( ( B +e -e 1 ) +e -e ( B +e -e 1 ) ) )
127	97, 95, 126	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) = ( ( B +e -e 1 ) +e -e ( B +e -e 1 ) ) )
128		xnegid 12069	. . . . . . . . 9 |- ( ( B +e -e 1 ) e. RR* -> ( ( B +e -e 1 ) +e -e ( B +e -e 1 ) ) = 0 )
129	95, 128	syl 17	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( ( B +e -e 1 ) +e -e ( B +e -e 1 ) ) = 0 )
130	127, 129	eqtrd 2656	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) = 0 )
131	130	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) ) = ( A +e 0 ) )
132	96, 6	syl 17	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e 0 ) = A )
133	131, 132	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e ( -e ( B +e -e 1 ) +e ( B +e -e 1 ) ) ) = A )
134	100, 125, 133	3eqtrrd 2661	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> A = ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) )
135	112	ltm1d 10956	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B - 1 ) < B )
136	114, 135	eqbrtrd 4675	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( B +e -e 1 ) < B )
137	101, 116	resubcld 10458	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A - ( B - 1 ) ) e. RR )
138		ltpnf 11954	. . . . . 6 |- ( ( A - ( B - 1 ) ) e. RR -> ( A - ( B - 1 ) ) < +oo )
139	137, 138	syl 17	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A - ( B - 1 ) ) < +oo )
140		rexsub 12064	. . . . . . 7 |- ( ( A e. RR /\ ( B +e -e 1 ) e. RR ) -> ( A +e -e ( B +e -e 1 ) ) = ( A - ( B +e -e 1 ) ) )
141	101, 117, 140	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e -e ( B +e -e 1 ) ) = ( A - ( B +e -e 1 ) ) )
142	114	oveq2d 6666	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A - ( B +e -e 1 ) ) = ( A - ( B - 1 ) ) )
143	141, 142	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e -e ( B +e -e 1 ) ) = ( A - ( B - 1 ) ) )
144		simpr 477	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> C = +oo )
145	139, 143, 144	3brtr4d 4685	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> ( A +e -e ( B +e -e 1 ) ) < C )
146		oveq1 6657	. . . . . . 7 |- ( b = ( B +e -e 1 ) -> ( b +e c ) = ( ( B +e -e 1 ) +e c ) )
147	146	eqeq2d 2632	. . . . . 6 |- ( b = ( B +e -e 1 ) -> ( A = ( b +e c ) <-> A = ( ( B +e -e 1 ) +e c ) ) )
148		breq1 4656	. . . . . 6 |- ( b = ( B +e -e 1 ) -> ( b < B <-> ( B +e -e 1 ) < B ) )
149	147, 148	3anbi12d 1400	. . . . 5 |- ( b = ( B +e -e 1 ) -> ( ( A = ( b +e c ) /\ b < B /\ c < C ) <-> ( A = ( ( B +e -e 1 ) +e c ) /\ ( B +e -e 1 ) < B /\ c < C ) ) )
150		oveq2 6658	. . . . . . 7 |- ( c = ( A +e -e ( B +e -e 1 ) ) -> ( ( B +e -e 1 ) +e c ) = ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) )
151	150	eqeq2d 2632	. . . . . 6 |- ( c = ( A +e -e ( B +e -e 1 ) ) -> ( A = ( ( B +e -e 1 ) +e c ) <-> A = ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) ) )
152		breq1 4656	. . . . . 6 |- ( c = ( A +e -e ( B +e -e 1 ) ) -> ( c < C <-> ( A +e -e ( B +e -e 1 ) ) < C ) )
153	151, 152	3anbi13d 1401	. . . . 5 |- ( c = ( A +e -e ( B +e -e 1 ) ) -> ( ( A = ( ( B +e -e 1 ) +e c ) /\ ( B +e -e 1 ) < B /\ c < C ) <-> ( A = ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) /\ ( B +e -e 1 ) < B /\ ( A +e -e ( B +e -e 1 ) ) < C ) ) )
154	149, 153	rspc2ev 3324	. . . 4 |- ( ( ( B +e -e 1 ) e. RR* /\ ( A +e -e ( B +e -e 1 ) ) e. RR* /\ ( A = ( ( B +e -e 1 ) +e ( A +e -e ( B +e -e 1 ) ) ) /\ ( B +e -e 1 ) < B /\ ( A +e -e ( B +e -e 1 ) ) < C ) ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
155	95, 98, 134, 136, 145, 154	syl113anc 1338	. . 3 |- ( ( ( ph /\ B =/= +oo ) /\ C = +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
156	1	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> A e. RR )
157	91	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> B e. RR* )
158		simplr 792	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> B =/= +oo )
159	157, 158, 105	syl2anc 693	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> ( B e. RR \/ B = -oo ) )
160	159	orcomd 403	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> ( B = -oo \/ B e. RR ) )
161	108	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> B =/= -oo )
162	161	neneqd 2799	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> -. B = -oo )
163	160, 162, 111	sylc 65	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> B e. RR )
164	28	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> C e. RR* )
165	164, 40	sylancom 701	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> ( C e. RR \/ C = -oo ) )
166	165	orcomd 403	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> ( C = -oo \/ C e. RR ) )
167	43	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> C =/= -oo )
168	167	neneqd 2799	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> -. C = -oo )
169	166, 168, 46	sylc 65	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> C e. RR )
170		xlt2addrd.6	. . . . . . . 8 |- ( ph -> A < ( B +e C ) )
171	170	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> A < ( B +e C ) )
172		rexadd 12063	. . . . . . . 8 |- ( ( B e. RR /\ C e. RR ) -> ( B +e C ) = ( B + C ) )
173	163, 169, 172	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> ( B +e C ) = ( B + C ) )
174	171, 173	breqtrd 4679	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> A < ( B + C ) )
175	156, 163, 169, 174	lt2addrd 29516	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )
176		rexadd 12063	. . . . . . . 8 |- ( ( b e. RR /\ c e. RR ) -> ( b +e c ) = ( b + c ) )
177	176	eqeq2d 2632	. . . . . . 7 |- ( ( b e. RR /\ c e. RR ) -> ( A = ( b +e c ) <-> A = ( b + c ) ) )
178	177	3anbi1d 1403	. . . . . 6 |- ( ( b e. RR /\ c e. RR ) -> ( ( A = ( b +e c ) /\ b < B /\ c < C ) <-> ( A = ( b + c ) /\ b < B /\ c < C ) ) )
179	178	2rexbiia 3055	. . . . 5 |- ( E. b e. RR E. c e. RR ( A = ( b +e c ) /\ b < B /\ c < C ) <-> E. b e. RR E. c e. RR ( A = ( b + c ) /\ b < B /\ c < C ) )
180	175, 179	sylibr 224	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> E. b e. RR E. c e. RR ( A = ( b +e c ) /\ b < B /\ c < C ) )
181		ressxr 10083	. . . . . 6 |- RR C_ RR*
182		ssrexv 3667	. . . . . 6 |- ( RR C_ RR* -> ( E. c e. RR ( A = ( b +e c ) /\ b < B /\ c < C ) -> E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) ) )
183	181, 182	ax-mp 5	. . . . 5 |- ( E. c e. RR ( A = ( b +e c ) /\ b < B /\ c < C ) -> E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
184	183	reximi 3011	. . . 4 |- ( E. b e. RR E. c e. RR ( A = ( b +e c ) /\ b < B /\ c < C ) -> E. b e. RR E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
185		ssrexv 3667	. . . . 5 |- ( RR C_ RR* -> ( E. b e. RR E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) ) )
186	181, 185	ax-mp 5	. . . 4 |- ( E. b e. RR E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
187	180, 184, 186	3syl 18	. . 3 |- ( ( ( ph /\ B =/= +oo ) /\ C =/= +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
188	155, 187	pm2.61dane 2881	. 2 |- ( ( ph /\ B =/= +oo ) -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )
189	90, 188	pm2.61dane 2881	1 |- ( ph -> E. b e. RR* E. c e. RR* ( A = ( b +e c ) /\ b < B /\ c < C ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   - cmin 10266   -ucneg 10267   -ecxne 11943   +ecxad 11944

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079  df-rp 11833  df-xneg 11946  df-xadd 11947

This theorem is referenced by:  xrofsup  29533