Theorem projlem10 9195

Description: Part of Lemma 3.6 of [Beran] p. 100 (lemma for projection theorem). A member of the infimum set. Used by projlem12 9197.

Hypotheses

Ref	Expression
projlem8.1	|- A e. H~
projlem8.2	|- H e. CH
projlem8.3	|- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}

Assertion

Ref	Expression
projlem10	|- (B e. H -> -u(normh` (B -h A)) e. S)

Distinct variable groups:   v,u,A   u,H,v

Proof of Theorem projlem10

Step	Hyp	Ref	Expression
1		opreq1 3968	. . . . 5 |- (x = B -> (x -h A) = (B -h A))
2	1	fveq2d 3728	. . . 4 |- (x = B -> (normh` (x -h A)) = (normh` (B -h A)))
3	2	negeqd 5361	. . 3 |- (x = B -> -u(normh` (x -h A)) = -u(normh` (B -h A)))
4	3	eleq1d 1540	. 2 |- (x = B -> (-u(normh` (x -h A)) e. S <-> -u(normh` (B -h A)) e. S))
5		projlem8.2	. . . . . . 7 |- H e. CH
6	5	chel 9102	. . . . . 6 |- (x e. H -> x e. H~)
7		projlem8.1	. . . . . . 7 |- A e. H~
8		hvsubclt 8887	. . . . . . 7 |- ((x e. H~ /\ A e. H~) -> (x -h A) e. H~)
9	7, 8	mpan2 696	. . . . . 6 |- (x e. H~ -> (x -h A) e. H~)
10	6, 9	syl 10	. . . . 5 |- (x e. H -> (x -h A) e. H~)
11		normclt 8991	. . . . 5 |- ((x -h A) e. H~ -> (normh` (x -h A)) e. RR)
12		renegclt 5437	. . . . 5 |- ((normh` (x -h A)) e. RR -> -u(normh` (x -h A)) e. RR)
13	10, 11, 12	3syl 20	. . . 4 |- (x e. H -> -u(normh` (x -h A)) e. RR)
14		eqid 1475	. . . . 5 |- -u(normh` (x -h A)) = -u(normh` (x -h A))
15		opreq1 3968	. . . . . . . . 9 |- (v = x -> (v -h A) = (x -h A))
16	15	fveq2d 3728	. . . . . . . 8 |- (v = x -> (normh` (v -h A)) = (normh` (x -h A)))
17	16	negeqd 5361	. . . . . . 7 |- (v = x -> -u(normh` (v -h A)) = -u(normh` (x -h A)))
18	17	eqeq2d 1486	. . . . . 6 |- (v = x -> (-u(normh` (x -h A)) = -u(normh` (v -h A)) <-> -u(normh` (x -h A)) = -u(normh` (x -h A))))
19	18	rcla4ev 1877	. . . . 5 |- ((x e. H /\ -u(normh` (x -h A)) = -u(normh` (x -h A))) -> E.v e. H -u(normh` (x -h A)) = -u(normh` (v -h A)))
20	14, 19	mpan2 696	. . . 4 |- (x e. H -> E.v e. H -u(normh` (x -h A)) = -u(normh` (v -h A)))
21	13, 20	jca 288	. . 3 |- (x e. H -> (-u(normh` (x -h A)) e. RR /\ E.v e. H -u(normh` (x -h A)) = -u(normh` (v -h A))))
22		eqeq1 1481	. . . . 5 |- (u = -u(normh` (x -h A)) -> (u = -u(normh` (v -h A)) <-> -u(normh` (x -h A)) = -u(normh` (v -h A))))
23	22	rexbidv 1664	. . . 4 |- (u = -u(normh` (x -h A)) -> (E.v e. H u = -u(normh` (v -h A)) <-> E.v e. H -u(normh` (x -h A)) = -u(normh` (v -h A))))
24		projlem8.3	. . . 4 |- S = {u e. RR | E.v e. H u = -u(normh` (v -h A))}
25	23, 24	elrab2 1907	. . 3 |- (-u(normh` (x -h A)) e. S <-> (-u(normh` (x -h A)) e. RR /\ E.v e. H -u(normh` (x -h A)) = -u(normh` (v -h A))))
26	21, 25	sylibr 200	. 2 |- (x e. H -> -u(normh` (x -h A)) e. S)
27	4, 26	vtoclga 1852	1 |- (B e. H -> -u(normh` (B -h A)) e. S)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wrex 1646  {crab 1648  ` cfv 3182  (class class class)co 3963  RRcr 5233  -ucneg 5293  H~chil 8788   -h cmv 8792  normhcno 8794  CHcch 8798

This theorem is referenced by:  projlem12 9197

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  ax-hilex 8869  ax-hfvadd 8870  ax-hv0cl 8873  ax-hfvmul 8875  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his3 8951  ax-his4 8952

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 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-hnorm 8837  df-hvsub 8840  df-sh 9076  df-ch 9092

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