Theorem minveclem35 8510

Description: Lemma for minveceu 8514.

Hypotheses

Ref	Expression
minvec35.x	|- X = (Base` U)
minvec35.g	|- G = (+v` U)
minvec35.m	|- M = (-v` U)
minvec35.s	|- S = (.s` U)
minvec35.n	|- N = (norm` U)
minvec35.y	|- Y = (Base` W)
minvec35.u	|- U e. CPreHil
minvec35.a	|- A e. X

Assertion

Ref	Expression
minveclem35	|- ((a e. X /\ b e. X) -> (N` ((AMa)G(AMb))) = (2 x. (N` (AM((1 / 2)S(aGb))))))

Distinct variable group:   a,b

Proof of Theorem minveclem35

Step	Hyp	Ref	Expression
1		minvec35.a	. . . . 5 |- A e. X
2		minvec35.u	. . . . . . 7 |- U e. CPreHil
3	2	phnvi 8406	. . . . . 6 |- U e. NrmCVec
4		minvec35.x	. . . . . . 7 |- X = (Base` U)
5		minvec35.g	. . . . . . 7 |- G = (+v` U)
6		minvec35.m	. . . . . . 7 |- M = (-v` U)
7	4, 5, 6	nvaddsub4 8221	. . . . . 6 |- ((U e. NrmCVec /\ (A e. X /\ A e. X) /\ (a e. X /\ b e. X)) -> ((AGA)M(aGb)) = ((AMa)G(AMb)))
8	3, 7	mp3an1 900	. . . . 5 |- (((A e. X /\ A e. X) /\ (a e. X /\ b e. X)) -> ((AGA)M(aGb)) = ((AMa)G(AMb)))
9	1, 1, 8	mpanl12 706	. . . 4 |- ((a e. X /\ b e. X) -> ((AGA)M(aGb)) = ((AMa)G(AMb)))
10	4, 5	nvgcl 8179	. . . . . 6 |- ((U e. NrmCVec /\ a e. X /\ b e. X) -> (aGb) e. X)
11	3, 10	mp3an1 900	. . . . 5 |- ((a e. X /\ b e. X) -> (aGb) e. X)
12		minvec35.s	. . . . . . . . . 10 |- S = (.s` U)
13	4, 5, 12	nv2 8193	. . . . . . . . 9 |- ((U e. NrmCVec /\ A e. X) -> (AGA) = (2SA))
14	3, 1, 13	mp2an 695	. . . . . . . 8 |- (AGA) = (2SA)
15	14	a1i 8	. . . . . . 7 |- ((aGb) e. X -> (AGA) = (2SA))
16		2cn 5927	. . . . . . . . . . 11 |- 2 e. CC
17		2ne0 5937	. . . . . . . . . . 11 |- 2 =/= 0
18	16, 17	recid 5696	. . . . . . . . . 10 |- (2 x. (1 / 2)) = 1
19	18	opreq1i 3956	. . . . . . . . 9 |- ((2 x. (1 / 2))S(aGb)) = (1S(aGb))
20	19	a1i 8	. . . . . . . 8 |- ((aGb) e. X -> ((2 x. (1 / 2))S(aGb)) = (1S(aGb)))
21	16, 17	reccl 5682	. . . . . . . . 9 |- (1 / 2) e. CC
22	4, 12	nvsass 8189	. . . . . . . . . 10 |- ((U e. NrmCVec /\ (2 e. CC /\ (1 / 2) e. CC /\ (aGb) e. X)) -> ((2 x. (1 / 2))S(aGb)) = (2S((1 / 2)S(aGb))))
23	3, 22	mpan 693	. . . . . . . . 9 |- ((2 e. CC /\ (1 / 2) e. CC /\ (aGb) e. X) -> ((2 x. (1 / 2))S(aGb)) = (2S((1 / 2)S(aGb))))
24	16, 21, 23	mp3an12 903	. . . . . . . 8 |- ((aGb) e. X -> ((2 x. (1 / 2))S(aGb)) = (2S((1 / 2)S(aGb))))
25	4, 12	nvsid 8188	. . . . . . . . 9 |- ((U e. NrmCVec /\ (aGb) e. X) -> (1S(aGb)) = (aGb))
26	3, 25	mpan 693	. . . . . . . 8 |- ((aGb) e. X -> (1S(aGb)) = (aGb))
27	20, 24, 26	3eqtr3rd 1508	. . . . . . 7 |- ((aGb) e. X -> (aGb) = (2S((1 / 2)S(aGb))))
28	15, 27	opreq12d 3963	. . . . . 6 |- ((aGb) e. X -> ((AGA)M(aGb)) = ((2SA)M(2S((1 / 2)S(aGb)))))
29	4, 12	nvscl 8187	. . . . . . . 8 |- ((U e. NrmCVec /\ (1 / 2) e. CC /\ (aGb) e. X) -> ((1 / 2)S(aGb)) e. X)
30	3, 21, 29	mp3an12 903	. . . . . . 7 |- ((aGb) e. X -> ((1 / 2)S(aGb)) e. X)
31	4, 6, 12	nvmdi 8210	. . . . . . . . 9 |- ((U e. NrmCVec /\ (2 e. CC /\ A e. X /\ ((1 / 2)S(aGb)) e. X)) -> (2S(AM((1 / 2)S(aGb)))) = ((2SA)M(2S((1 / 2)S(aGb)))))
32	3, 31	mpan 693	. . . . . . . 8 |- ((2 e. CC /\ A e. X /\ ((1 / 2)S(aGb)) e. X) -> (2S(AM((1 / 2)S(aGb)))) = ((2SA)M(2S((1 / 2)S(aGb)))))
33	16, 1, 32	mp3an12 903	. . . . . . 7 |- (((1 / 2)S(aGb)) e. X -> (2S(AM((1 / 2)S(aGb)))) = ((2SA)M(2S((1 / 2)S(aGb)))))
34	30, 33	syl 10	. . . . . 6 |- ((aGb) e. X -> (2S(AM((1 / 2)S(aGb)))) = ((2SA)M(2S((1 / 2)S(aGb)))))
35	28, 34	eqtr4d 1502	. . . . 5 |- ((aGb) e. X -> ((AGA)M(aGb)) = (2S(AM((1 / 2)S(aGb)))))
36	11, 35	syl 10	. . . 4 |- ((a e. X /\ b e. X) -> ((AGA)M(aGb)) = (2S(AM((1 / 2)S(aGb)))))
37	9, 36	eqtr3d 1501	. . 3 |- ((a e. X /\ b e. X) -> ((AMa)G(AMb)) = (2S(AM((1 / 2)S(aGb)))))
38	37	fveq2d 3713	. 2 |- ((a e. X /\ b e. X) -> (N` ((AMa)G(AMb))) = (N` (2S(AM((1 / 2)S(aGb))))))
39	11, 30	syl 10	. . . 4 |- ((a e. X /\ b e. X) -> ((1 / 2)S(aGb)) e. X)
40	4, 6	nvmcl 8207	. . . . 5 |- ((U e. NrmCVec /\ A e. X /\ ((1 / 2)S(aGb)) e. X) -> (AM((1 / 2)S(aGb))) e. X)
41	3, 1, 40	mp3an12 903	. . . 4 |- (((1 / 2)S(aGb)) e. X -> (AM((1 / 2)S(aGb))) e. X)
42	39, 41	syl 10	. . 3 |- ((a e. X /\ b e. X) -> (AM((1 / 2)S(aGb))) e. X)
43		2re 5926	. . . 4 |- 2 e. RR
44		0re 5412	. . . . 5 |- 0 e. RR
45		2pos 5936	. . . . 5 |- 0 < 2
46	44, 43, 45	ltlei 5554	. . . 4 |- 0 <_ 2
47		minvec35.n	. . . . . 6 |- N = (norm` U)
48	4, 12, 47	nvsge0 8230	. . . . 5 |- ((U e. NrmCVec /\ (2 e. RR /\ 0 <_ 2) /\ (AM((1 / 2)S(aGb))) e. X) -> (N` (2S(AM((1 / 2)S(aGb))))) = (2 x. (N` (AM((1 / 2)S(aGb))))))
49	3, 48	mp3an1 900	. . . 4 |- (((2 e. RR /\ 0 <_ 2) /\ (AM((1 / 2)S(aGb))) e. X) -> (N` (2S(AM((1 / 2)S(aGb))))) = (2 x. (N` (AM((1 / 2)S(aGb))))))
50	43, 46, 49	mpanl12 706	. . 3 |- ((AM((1 / 2)S(aGb))) e. X -> (N` (2S(AM((1 / 2)S(aGb))))) = (2 x. (N` (AM((1 / 2)S(aGb))))))
51	42, 50	syl 10	. 2 |- ((a e. X /\ b e. X) -> (N` (2S(AM((1 / 2)S(aGb))))) = (2 x. (N` (AM((1 / 2)S(aGb))))))
52	38, 51	eqtrd 1499	1 |- ((a e. X /\ b e. X) -> (N` ((AMa)G(AMb))) = (2 x. (N` (AM((1 / 2)S(aGb))))))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204  RRcr 5205  0cc0 5206  1c1 5207   x. cmul 5211   / cdiv 5266   <_ cle 5267  2c2 5908  NrmCVeccnv 8141  +vcpv 8142  Basecba 8143  .scns 8144  -vcnsb 8146  normcnm 8147  CPreHilcphl 8402

This theorem is referenced by:  minveclem38 8513

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