Theorem minveclem35 8579

Description: Lemma for minveceu 8583.

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 8475	. . . . . 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 8281	. . . . . 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 903	. . . . 5 |- (((A e. X /\ A e. X) /\ (a e. X /\ b e. X)) -> ((AGA)M(aGb)) = ((AMa)G(AMb)))
9	1, 1, 8	mpanl12 708	. . . 4 |- ((a e. X /\ b e. X) -> ((AGA)M(aGb)) = ((AMa)G(AMb)))
10	4, 5	nvgcl 8239	. . . . . 6 |- ((U e. NrmCVec /\ a e. X /\ b e. X) -> (aGb) e. X)
11	3, 10	mp3an1 903	. . . . 5 |- ((a e. X /\ b e. X) -> (aGb) e. X)
12		minvec35.s	. . . . . . . . . 10 |- S = (.s` U)
13	4, 5, 12	nv2 8253	. . . . . . . . 9 |- ((U e. NrmCVec /\ A e. X) -> (AGA) = (2SA))
14	3, 1, 13	mp2an 697	. . . . . . . 8 |- (AGA) = (2SA)
15	14	a1i 8	. . . . . . 7 |- ((aGb) e. X -> (AGA) = (2SA))
16		2cn 5980	. . . . . . . . . . 11 |- 2 e. CC
17		2ne0 5990	. . . . . . . . . . 11 |- 2 =/= 0
18	16, 17	recid 5733	. . . . . . . . . 10 |- (2 x. (1 / 2)) = 1
19	18	opreq1i 3971	. . . . . . . . 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 5713	. . . . . . . . 9 |- (1 / 2) e. CC
22	4, 12	nvsass 8249	. . . . . . . . . 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 695	. . . . . . . . 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 906	. . . . . . . 8 |- ((aGb) e. X -> ((2 x. (1 / 2))S(aGb)) = (2S((1 / 2)S(aGb))))
25	4, 12	nvsid 8248	. . . . . . . . 9 |- ((U e. NrmCVec /\ (aGb) e. X) -> (1S(aGb)) = (aGb))
26	3, 25	mpan 695	. . . . . . . 8 |- ((aGb) e. X -> (1S(aGb)) = (aGb))
27	20, 24, 26	3eqtr3rd 1516	. . . . . . 7 |- ((aGb) e. X -> (aGb) = (2S((1 / 2)S(aGb))))
28	15, 27	opreq12d 3978	. . . . . 6 |- ((aGb) e. X -> ((AGA)M(aGb)) = ((2SA)M(2S((1 / 2)S(aGb)))))
29	4, 12	nvscl 8247	. . . . . . . 8 |- ((U e. NrmCVec /\ (1 / 2) e. CC /\ (aGb) e. X) -> ((1 / 2)S(aGb)) e. X)
30	3, 21, 29	mp3an12 906	. . . . . . 7 |- ((aGb) e. X -> ((1 / 2)S(aGb)) e. X)
31	4, 6, 12	nvmdi 8270	. . . . . . . . 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 695	. . . . . . . 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 906	. . . . . . 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 1510	. . . . 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 1509	. . 3 |- ((a e. X /\ b e. X) -> ((AMa)G(AMb)) = (2S(AM((1 / 2)S(aGb)))))
38	37	fveq2d 3728	. 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 8267	. . . . 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 906	. . . 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 5979	. . . 4 |- 2 e. RR
44		0re 5440	. . . . 5 |- 0 e. RR
45		2pos 5989	. . . . 5 |- 0 < 2
46	44, 43, 45	ltlei 5581	. . . 4 |- 0 <_ 2
47		minvec35.n	. . . . . 6 |- N = (norm` U)
48	4, 12, 47	nvsge0 8291	. . . . 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 903	. . . 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 708	. . 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 1507	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 775   = wceq 956   e. wcel 958   class class class wbr 2619  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234  1c1 5235   x. cmul 5239   / cdiv 5294   <_ cle 5295  2c2 5961  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  -vcnsb 8208  normcnm 8209  CPreHilcphl 8471

This theorem is referenced by:  minveclem38 8582

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