Theorem truni1 10386

Description: Translation in a half-infinite interval.

Assertion

Ref	Expression
truni1	|- ((A e. RR* /\ D e. RR /\ 0 < D) -> (C e. (A(,) +oo) -> (C + D) e. (A(,) +oo)))

Proof of Theorem truni1

Step	Hyp	Ref	Expression
1		axaddrcl 5244	. . . . . . . . 9 |- ((C e. RR /\ D e. RR) -> (C + D) e. RR)
2	1	ex 373	. . . . . . . 8 |- (C e. RR -> (D e. RR -> (C + D) e. RR))
3	2	3ad2ant1 798	. . . . . . 7 |- ((C e. RR /\ A < C /\ C < +oo) -> (D e. RR -> (C + D) e. RR))
4	3	com12 11	. . . . . 6 |- (D e. RR -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) e. RR))
5	4	3ad2ant2 799	. . . . 5 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) e. RR))
6	5	imp 350	. . . 4 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> (C + D) e. RR)
7		xrlttrt 5526	. . . . 5 |- ((A e. RR* /\ C e. RR* /\ (C + D) e. RR*) -> ((A < C /\ C < (C + D)) -> A < (C + D)))
8		3simp1 786	. . . . . . 7 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> A e. RR*)
9	8	adantr 389	. . . . . 6 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> A e. RR*)
10		rexrt 5471	. . . . . . . 8 |- (C e. RR -> C e. RR*)
11	10	3ad2ant1 798	. . . . . . 7 |- ((C e. RR /\ A < C /\ C < +oo) -> C e. RR*)
12	11	adantl 388	. . . . . 6 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> C e. RR*)
13		rexrt 5471	. . . . . . . . . . . 12 |- ((C + D) e. RR -> (C + D) e. RR*)
14	1, 13	syl 10	. . . . . . . . . . 11 |- ((C e. RR /\ D e. RR) -> (C + D) e. RR*)
15	14	ex 373	. . . . . . . . . 10 |- (C e. RR -> (D e. RR -> (C + D) e. RR*))
16	15	3ad2ant1 798	. . . . . . . . 9 |- ((C e. RR /\ A < C /\ C < +oo) -> (D e. RR -> (C + D) e. RR*))
17	16	com12 11	. . . . . . . 8 |- (D e. RR -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) e. RR*))
18	17	3ad2ant2 799	. . . . . . 7 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) e. RR*))
19	18	imp 350	. . . . . 6 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> (C + D) e. RR*)
20	9, 12, 19	3jca 817	. . . . 5 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> (A e. RR* /\ C e. RR* /\ (C + D) e. RR*))
21		3simp2 787	. . . . . . 7 |- ((C e. RR /\ A < C /\ C < +oo) -> A < C)
22	21	adantl 388	. . . . . 6 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> A < C)
23		ltaddpos2t 5625	. . . . . . . . . . . . . . . 16 |- ((D e. RR /\ C e. RR) -> (0 < D <-> C < (D + C)))
24	23	biimpd 153	. . . . . . . . . . . . . . 15 |- ((D e. RR /\ C e. RR) -> (0 < D -> C < (D + C)))
25	24	ex 373	. . . . . . . . . . . . . 14 |- (D e. RR -> (C e. RR -> (0 < D -> C < (D + C))))
26	25	com23 32	. . . . . . . . . . . . 13 |- (D e. RR -> (0 < D -> (C e. RR -> C < (D + C))))
27	26	imp31 362	. . . . . . . . . . . 12 |- (((D e. RR /\ 0 < D) /\ C e. RR) -> C < (D + C))
28		axaddcom 5247	. . . . . . . . . . . . 13 |- ((C e. CC /\ D e. CC) -> (C + D) = (D + C))
29		recnt 5285	. . . . . . . . . . . . . 14 |- (C e. RR -> C e. CC)
30	29	adantl 388	. . . . . . . . . . . . 13 |- (((D e. RR /\ 0 < D) /\ C e. RR) -> C e. CC)
31		recnt 5285	. . . . . . . . . . . . . 14 |- (D e. RR -> D e. CC)
32	31	ad2antrr 404	. . . . . . . . . . . . 13 |- (((D e. RR /\ 0 < D) /\ C e. RR) -> D e. CC)
33	28, 30, 32	sylanc 471	. . . . . . . . . . . 12 |- (((D e. RR /\ 0 < D) /\ C e. RR) -> (C + D) = (D + C))
34	27, 33	breqtrrd 2631	. . . . . . . . . . 11 |- (((D e. RR /\ 0 < D) /\ C e. RR) -> C < (C + D))
35	34	expcom 374	. . . . . . . . . 10 |- (C e. RR -> ((D e. RR /\ 0 < D) -> C < (C + D)))
36	35	3ad2ant1 798	. . . . . . . . 9 |- ((C e. RR /\ A < C /\ C < +oo) -> ((D e. RR /\ 0 < D) -> C < (C + D)))
37	36	com12 11	. . . . . . . 8 |- ((D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> C < (C + D)))
38	37	3adant1 795	. . . . . . 7 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> C < (C + D)))
39	38	imp 350	. . . . . 6 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> C < (C + D))
40	22, 39	jca 288	. . . . 5 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> (A < C /\ C < (C + D)))
41	7, 20, 40	sylc 68	. . . 4 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> A < (C + D))
42		ltpnft 5515	. . . . . . . . . 10 |- ((C + D) e. RR -> (C + D) < +oo)
43	1, 42	syl 10	. . . . . . . . 9 |- ((C e. RR /\ D e. RR) -> (C + D) < +oo)
44	43	ex 373	. . . . . . . 8 |- (C e. RR -> (D e. RR -> (C + D) < +oo))
45	44	3ad2ant1 798	. . . . . . 7 |- ((C e. RR /\ A < C /\ C < +oo) -> (D e. RR -> (C + D) < +oo))
46	45	com12 11	. . . . . 6 |- (D e. RR -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) < +oo))
47	46	3ad2ant2 799	. . . . 5 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> (C + D) < +oo))
48	47	imp 350	. . . 4 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> (C + D) < +oo)
49	6, 41, 48	3jca 817	. . 3 |- (((A e. RR* /\ D e. RR /\ 0 < D) /\ (C e. RR /\ A < C /\ C < +oo)) -> ((C + D) e. RR /\ A < (C + D) /\ (C + D) < +oo))
50	49	ex 373	. 2 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C e. RR /\ A < C /\ C < +oo) -> ((C + D) e. RR /\ A < (C + D) /\ (C + D) < +oo)))
51		pnfxr 5465	. . . 4 |- +oo e. RR*
52	8, 51	jctir 293	. . 3 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> (A e. RR* /\ +oo e. RR*))
53		elioo2t 6316	. . 3 |- ((A e. RR* /\ +oo e. RR*) -> (C e. (A(,) +oo) <-> (C e. RR /\ A < C /\ C < +oo)))
54	52, 53	syl 10	. 2 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> (C e. (A(,) +oo) <-> (C e. RR /\ A < C /\ C < +oo)))
55		elioo2t 6316	. . 3 |- ((A e. RR* /\ +oo e. RR*) -> ((C + D) e. (A(,) +oo) <-> ((C + D) e. RR /\ A < (C + D) /\ (C + D) < +oo)))
56	52, 55	syl 10	. 2 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> ((C + D) e. (A(,) +oo) <-> ((C + D) e. RR /\ A < (C + D) /\ (C + D) < +oo)))
57	50, 54, 56	3imtr4d 541	1 |- ((A e. RR* /\ D e. RR /\ 0 < D) -> (C e. (A(,) +oo) -> (C + D) e. (A(,) +oo)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206   + caddc 5209   +oocpnf 5455  RR*cxr 5457   < clt 5458  (,)cioo 6294

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni