Theorem truni1 10499

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 5272	. . . . . . . . 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 800	. . . . . . 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 801	. . . . 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 5553	. . . . 5 |- ((A e. RR* /\ C e. RR* /\ (C + D) e. RR*) -> ((A < C /\ C < (C + D)) -> A < (C + D)))
8		3simp1 788	. . . . . . 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 5499	. . . . . . . 8 |- (C e. RR -> C e. RR*)
11	10	3ad2ant1 800	. . . . . . 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 5499	. . . . . . . . . . . 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 800	. . . . . . . . 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 801	. . . . . . 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 819	. . . . 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 789	. . . . . . 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 5652	. . . . . . . . . . . . . . . 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 5275	. . . . . . . . . . . . 13 |- ((C e. CC /\ D e. CC) -> (C + D) = (D + C))
29		recnt 5313	. . . . . . . . . . . . . 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 5313	. . . . . . . . . . . . . 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 2641	. . . . . . . . . . 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 800	. . . . . . . . 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 797	. . . . . . 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 5542	. . . . . . . . . 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 800	. . . . . . 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 801	. . . . 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 819	. . 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 5493	. . . 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 6379	. . 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 6379	. . 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 543	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 775   = wceq 956   e. wcel 958   class class class wbr 2619  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   + caddc 5237   +oocpnf 5483  RR*cxr 5485   < clt 5486  (,)cioo 6357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom