Theorem xrlexaddrp 39568

Description: If an extended real number A can be approximated from above, adding positive reals to B, then A is smaller or equal than B. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
xrlexaddrp.1	|- ( ph -> A e. RR* )
xrlexaddrp.2	|- ( ph -> B e. RR* )
xrlexaddrp.3	|- ( ( ph /\ x e. RR+ ) -> A <_ ( B +e x ) )

Assertion

Ref	Expression
xrlexaddrp	|- ( ph -> A <_ B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem xrlexaddrp

Step	Hyp	Ref	Expression
1		xrlexaddrp.1	. . . . 5 |- ( ph -> A e. RR* )
2		pnfge 11964	. . . . 5 |- ( A e. RR* -> A <_ +oo )
3	1, 2	syl 17	. . . 4 |- ( ph -> A <_ +oo )
4	3	adantr 481	. . 3 |- ( ( ph /\ B = +oo ) -> A <_ +oo )
5		id 22	. . . . 5 |- ( B = +oo -> B = +oo )
6	5	eqcomd 2628	. . . 4 |- ( B = +oo -> +oo = B )
7	6	adantl 482	. . 3 |- ( ( ph /\ B = +oo ) -> +oo = B )
8	4, 7	breqtrd 4679	. 2 |- ( ( ph /\ B = +oo ) -> A <_ B )
9		simpl 473	. . 3 |- ( ( ph /\ -. B = +oo ) -> ph )
10		neqne 2802	. . . 4 |- ( -. B = +oo -> B =/= +oo )
11	10	adantl 482	. . 3 |- ( ( ph /\ -. B = +oo ) -> B =/= +oo )
12		simpr 477	. . . . . 6 |- ( ( ph /\ A = -oo ) -> A = -oo )
13		xrlexaddrp.2	. . . . . . . 8 |- ( ph -> B e. RR* )
14		mnfle 11969	. . . . . . . 8 |- ( B e. RR* -> -oo <_ B )
15	13, 14	syl 17	. . . . . . 7 |- ( ph -> -oo <_ B )
16	15	adantr 481	. . . . . 6 |- ( ( ph /\ A = -oo ) -> -oo <_ B )
17	12, 16	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ A = -oo ) -> A <_ B )
18	17	adantlr 751	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ A = -oo ) -> A <_ B )
19		simpl 473	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ -. A = -oo ) -> ( ph /\ B =/= +oo ) )
20		neqne 2802	. . . . . 6 |- ( -. A = -oo -> A =/= -oo )
21	20	adantl 482	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ -. A = -oo ) -> A =/= -oo )
22		simpll 790	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) -> ph )
23	13	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ B =/= +oo ) -> B e. RR* )
24		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ B =/= +oo ) -> B =/= +oo )
25	23, 24	jca 554	. . . . . . . . . . 11 |- ( ( ph /\ B =/= +oo ) -> ( B e. RR* /\ B =/= +oo ) )
26		xrnepnf 11952	. . . . . . . . . . 11 |- ( ( B e. RR* /\ B =/= +oo ) <-> ( B e. RR \/ B = -oo ) )
27	25, 26	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ B =/= +oo ) -> ( B e. RR \/ B = -oo ) )
28	27	adantr 481	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ -. B e. RR ) -> ( B e. RR \/ B = -oo ) )
29		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ B =/= +oo ) /\ -. B e. RR ) -> -. B e. RR )
30		pm2.53 388	. . . . . . . . 9 |- ( ( B e. RR \/ B = -oo ) -> ( -. B e. RR -> B = -oo ) )
31	28, 29, 30	sylc 65	. . . . . . . 8 |- ( ( ( ph /\ B =/= +oo ) /\ -. B e. RR ) -> B = -oo )
32	31	adantlr 751	. . . . . . 7 |- ( ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) /\ -. B e. RR ) -> B = -oo )
33		id 22	. . . . . . . . . . . . 13 |- ( ph -> ph )
34		1rp 11836	. . . . . . . . . . . . . 14 |- 1 e. RR+
35	34	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> 1 e. RR+ )
36		1re 10039	. . . . . . . . . . . . . . 15 |- 1 e. RR
37	36	elexi 3213	. . . . . . . . . . . . . 14 |- 1 e. _V
38		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( x e. RR+ <-> 1 e. RR+ ) )
39	38	anbi2d 740	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( ( ph /\ x e. RR+ ) <-> ( ph /\ 1 e. RR+ ) ) )
40		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( x = 1 -> ( B +e x ) = ( B +e 1 ) )
41	40	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( x = 1 -> ( A <_ ( B +e x ) <-> A <_ ( B +e 1 ) ) )
42	39, 41	imbi12d 334	. . . . . . . . . . . . . 14 |- ( x = 1 -> ( ( ( ph /\ x e. RR+ ) -> A <_ ( B +e x ) ) <-> ( ( ph /\ 1 e. RR+ ) -> A <_ ( B +e 1 ) ) ) )
43		xrlexaddrp.3	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. RR+ ) -> A <_ ( B +e x ) )
44	37, 42, 43	vtocl 3259	. . . . . . . . . . . . 13 |- ( ( ph /\ 1 e. RR+ ) -> A <_ ( B +e 1 ) )
45	33, 35, 44	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> A <_ ( B +e 1 ) )
46	45	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> A <_ ( B +e 1 ) )
47		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( B = -oo -> ( B +e 1 ) = ( -oo +e 1 ) )
48	36	rexri 10097	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR*
49		ltpnf 11954	. . . . . . . . . . . . . . . . . . . 20 |- ( 1 e. RR -> 1 < +oo )
50	36, 49	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 |- 1 < +oo
51	36, 50	ltneii 10150	. . . . . . . . . . . . . . . . . 18 |- 1 =/= +oo
52		xaddmnf2 12060	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. RR* /\ 1 =/= +oo ) -> ( -oo +e 1 ) = -oo )
53	48, 51, 52	mp2an 708	. . . . . . . . . . . . . . . . 17 |- ( -oo +e 1 ) = -oo
54	53	a1i 11	. . . . . . . . . . . . . . . 16 |- ( B = -oo -> ( -oo +e 1 ) = -oo )
55	47, 54	eqtr2d 2657	. . . . . . . . . . . . . . 15 |- ( B = -oo -> -oo = ( B +e 1 ) )
56	55	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> -oo = ( B +e 1 ) )
57	56	eqcomd 2628	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> ( B +e 1 ) = -oo )
58	1	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A =/= -oo ) -> A e. RR* )
59		simpr 477	. . . . . . . . . . . . . . 15 |- ( ( ph /\ A =/= -oo ) -> A =/= -oo )
60		nemnftgtmnft 39560	. . . . . . . . . . . . . . 15 |- ( ( A e. RR* /\ A =/= -oo ) -> -oo < A )
61	58, 59, 60	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ A =/= -oo ) -> -oo < A )
62	61	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> -oo < A )
63	57, 62	eqbrtrd 4675	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> ( B +e 1 ) < A )
64	13	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> B e. RR* )
65	48	a1i 11	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> 1 e. RR* )
66	64, 65	xaddcld 12131	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> ( B +e 1 ) e. RR* )
67	1	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> A e. RR* )
68		xrltnle 10105	. . . . . . . . . . . . 13 |- ( ( ( B +e 1 ) e. RR* /\ A e. RR* ) -> ( ( B +e 1 ) < A <-> -. A <_ ( B +e 1 ) ) )
69	66, 67, 68	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> ( ( B +e 1 ) < A <-> -. A <_ ( B +e 1 ) ) )
70	63, 69	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ph /\ A =/= -oo ) /\ B = -oo ) -> -. A <_ ( B +e 1 ) )
71	46, 70	pm2.65da 600	. . . . . . . . . 10 |- ( ( ph /\ A =/= -oo ) -> -. B = -oo )
72	71	neqned 2801	. . . . . . . . 9 |- ( ( ph /\ A =/= -oo ) -> B =/= -oo )
73	72	ad4ant13 1292	. . . . . . . 8 |- ( ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) /\ -. B e. RR ) -> B =/= -oo )
74	73	neneqd 2799	. . . . . . 7 |- ( ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) /\ -. B e. RR ) -> -. B = -oo )
75	32, 74	condan 835	. . . . . 6 |- ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) -> B e. RR )
76	43	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ B e. RR ) /\ x e. RR+ ) -> A <_ ( B +e x ) )
77		simpl 473	. . . . . . . . . . 11 |- ( ( B e. RR /\ x e. RR+ ) -> B e. RR )
78		rpre 11839	. . . . . . . . . . . 12 |- ( x e. RR+ -> x e. RR )
79	78	adantl 482	. . . . . . . . . . 11 |- ( ( B e. RR /\ x e. RR+ ) -> x e. RR )
80		rexadd 12063	. . . . . . . . . . 11 |- ( ( B e. RR /\ x e. RR ) -> ( B +e x ) = ( B + x ) )
81	77, 79, 80	syl2anc 693	. . . . . . . . . 10 |- ( ( B e. RR /\ x e. RR+ ) -> ( B +e x ) = ( B + x ) )
82	81	adantll 750	. . . . . . . . 9 |- ( ( ( ph /\ B e. RR ) /\ x e. RR+ ) -> ( B +e x ) = ( B + x ) )
83	76, 82	breqtrd 4679	. . . . . . . 8 |- ( ( ( ph /\ B e. RR ) /\ x e. RR+ ) -> A <_ ( B + x ) )
84	83	ralrimiva 2966	. . . . . . 7 |- ( ( ph /\ B e. RR ) -> A. x e. RR+ A <_ ( B + x ) )
85	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ B e. RR ) -> A e. RR* )
86		simpr 477	. . . . . . . 8 |- ( ( ph /\ B e. RR ) -> B e. RR )
87		xralrple 12036	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR ) -> ( A <_ B <-> A. x e. RR+ A <_ ( B + x ) ) )
88	85, 86, 87	syl2anc 693	. . . . . . 7 |- ( ( ph /\ B e. RR ) -> ( A <_ B <-> A. x e. RR+ A <_ ( B + x ) ) )
89	84, 88	mpbird 247	. . . . . 6 |- ( ( ph /\ B e. RR ) -> A <_ B )
90	22, 75, 89	syl2anc 693	. . . . 5 |- ( ( ( ph /\ B =/= +oo ) /\ A =/= -oo ) -> A <_ B )
91	19, 21, 90	syl2anc 693	. . . 4 |- ( ( ( ph /\ B =/= +oo ) /\ -. A = -oo ) -> A <_ B )
92	18, 91	pm2.61dan 832	. . 3 |- ( ( ph /\ B =/= +oo ) -> A <_ B )
93	9, 11, 92	syl2anc 693	. 2 |- ( ( ph /\ -. B = +oo ) -> A <_ B )
94	8, 93	pm2.61dan 832	1 |- ( ph -> A <_ B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   class class class wbr 4653  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   RR+crp 11832   +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  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-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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947

This theorem is referenced by:  infleinf  39588  sge0xaddlem2  40651  ovnsubadd  40786