Theorem pjthlem11 9224

Description: Lemma for pjth 9228.

Hypotheses

Ref	Expression
pjthlem9.1	|- A e. H~
pjthlem9.2	|- B e. H~
pjthlem9.3	|- D e. H~
pjthlem9.4	|- R = (1 / (D .ih D))
pjthlem9.5	|- S = (R x. (C .ih D))
pjthlem9.6	|- C = (A -h B)

Assertion

Ref	Expression
pjthlem11	|- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (C .ih D) = 0)

Proof of Theorem pjthlem11

Step	Hyp	Ref	Expression
1		pjthlem9.3	. . . . . . 7 |- D e. H~
2		pjthlem9.4	. . . . . . 7 |- R = (1 / (D .ih D))
3	1, 2	pjthlem3 9216	. . . . . 6 |- (D =/= 0h -> 0 < R)
4	1, 2	pjthlem2 9215	. . . . . . . . 9 |- (D =/= 0h -> R e. RR)
5		0re 5452	. . . . . . . . 9 |- 0 e. RR
6	4, 5	jctir 293	. . . . . . . 8 |- (D =/= 0h -> (R e. RR /\ 0 e. RR))
7		lttri2t 5525	. . . . . . . . 9 |- ((R e. RR /\ 0 e. RR) -> (R =/= 0 <-> (R < 0 \/ 0 < R)))
8		df-ne 1590	. . . . . . . . 9 |- (R =/= 0 <-> -. R = 0)
9	7, 8	syl5bbr 536	. . . . . . . 8 |- ((R e. RR /\ 0 e. RR) -> (-. R = 0 <-> (R < 0 \/ 0 < R)))
10	6, 9	syl 10	. . . . . . 7 |- (D =/= 0h -> (-. R = 0 <-> (R < 0 \/ 0 < R)))
11		olc 268	. . . . . . 7 |- (0 < R -> (R < 0 \/ 0 < R))
12	10, 11	syl5bir 210	. . . . . 6 |- (D =/= 0h -> (0 < R -> -. R = 0))
13	3, 12	mpd 26	. . . . 5 |- (D =/= 0h -> -. R = 0)
14	13	adantr 391	. . . 4 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> -. R = 0)
15		pjthlem9.1	. . . . . . 7 |- A e. H~
16		pjthlem9.2	. . . . . . 7 |- B e. H~
17		pjthlem9.5	. . . . . . 7 |- S = (R x. (C .ih D))
18		pjthlem9.6	. . . . . . 7 |- C = (A -h B)
19	15, 16, 1, 2, 17, 18	pjthlem10 9223	. . . . . 6 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (R x. ((abs` (C .ih D))^2)) = 0)
20	4	recnd 5327	. . . . . . . 8 |- (D =/= 0h -> R e. CC)
21	15, 16	hvsubcl 8886	. . . . . . . . . . . . . 14 |- (A -h B) e. H~
22	18, 21	eqeltr 1547	. . . . . . . . . . . . 13 |- C e. H~
23	22, 1	hicl 8943	. . . . . . . . . . . 12 |- (C .ih D) e. CC
24	23	abscl 6839	. . . . . . . . . . 11 |- (abs` (C .ih D)) e. RR
25	24	resqcl 6624	. . . . . . . . . 10 |- ((abs` (C .ih D))^2) e. RR
26	25	recn 5326	. . . . . . . . 9 |- ((abs` (C .ih D))^2) e. CC
27		mul0ort 5708	. . . . . . . . 9 |- ((R e. CC /\ ((abs` (C .ih D))^2) e. CC) -> ((R x. ((abs` (C .ih D))^2)) = 0 <-> (R = 0 \/ ((abs` (C .ih D))^2) = 0)))
28	26, 27	mpan2 698	. . . . . . . 8 |- (R e. CC -> ((R x. ((abs` (C .ih D))^2)) = 0 <-> (R = 0 \/ ((abs` (C .ih D))^2) = 0)))
29	20, 28	syl 10	. . . . . . 7 |- (D =/= 0h -> ((R x. ((abs` (C .ih D))^2)) = 0 <-> (R = 0 \/ ((abs` (C .ih D))^2) = 0)))
30	29	adantr 391	. . . . . 6 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((R x. ((abs` (C .ih D))^2)) = 0 <-> (R = 0 \/ ((abs` (C .ih D))^2) = 0)))
31	19, 30	mpbid 195	. . . . 5 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (R = 0 \/ ((abs` (C .ih D))^2) = 0))
32	31	ord 232	. . . 4 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (-. R = 0 -> ((abs` (C .ih D))^2) = 0))
33	14, 32	mpd 26	. . 3 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> ((abs` (C .ih D))^2) = 0)
34	24	recn 5326	. . . 4 |- (abs` (C .ih D)) e. CC
35	34	sqeq0 6617	. . 3 |- (((abs` (C .ih D))^2) = 0 <-> (abs` (C .ih D)) = 0)
36	33, 35	sylib 198	. 2 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (abs` (C .ih D)) = 0)
37	23	abs00 6842	. 2 |- ((abs` (C .ih D)) = 0 <-> (C .ih D) = 0)
38	36, 37	sylib 198	1 |- ((D =/= 0h /\ (normh` (B -h A)) <_ (normh` ((B +h (S .h D)) -h A))) -> (C .ih D) = 0)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588   class class class wbr 2624  ` cfv 3188  (class class class)co 3969  CCcc 5244  RRcr 5245  0cc0 5246  1c1 5247   x. cmul 5251   / cdiv 5306   <_ cle 5307   < clt 5498  2c2 5963  ^cexp 6569  abscabs 6751  H~chil 8783   +h cva 8784   .h csm 8785  0hc0v 8786   -h cmv 8787   .ih csp 8788  normhcno 8789

This theorem is referenced by:  pjthlem12 9225

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570  df-sqr 6671  df-re 6752  df-im 6753  df-cj 6754  df-abs 6755  df-hnorm 8832  df-hvsub 8835

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