Theorem sqrlem12 4742

Description: Lemma for square root theorem.

Hypotheses

Ref	Expression
sqrlem1.1	|- A e. RR
sqrlem1.2	|- 0 < A
sqrlem9.3	|- B e. RR
sqrlem9.4	|- C e. RR
sqrlem9.5	|- 0 < B
sqrlem9.6	|- A < (B x. B)
sqrlem9.7	|- C = ((B + (A / B)) / (1 + 1))
sqrlem12.8	|- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}

Assertion

Ref	Expression
sqrlem12	|- (D e. S -> D < C)

Distinct variable group(s):   x,A   x,S   x,D

Proof of Theorem sqrlem12

Step	Hyp	Ref	Expression
1		sqrlem1.1	. . . . . 6 |- A e. RR
2		sqrlem1.2	. . . . . 6 |- 0 < A
3		sqrlem12.8	. . . . . 6 |- S = {x e. RR | (0 <_ x /\ (x x. x) <_ A)}
4	1, 2, 3	sqrlem4 4734	. . . . 5 |- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))
5	4	pm3.27bd 263	. . . 4 |- (D e. S -> (0 <_ D /\ (D x. D) <_ A))
6	5	pm3.26d 258	. . 3 |- (D e. S -> 0 <_ D)
7	4	pm3.26bd 259	. . . 4 |- (D e. S -> D e. RR)
8		ax0re 4063	. . . . 5 |- 0 e. RR
9		leloet 4284	. . . . 5 |- ((0 e. RR /\ D e. RR) -> (0 <_ D <-> (0 < D \/ 0 = D)))
10	8, 9	mpan 518	. . . 4 |- (D e. RR -> (0 <_ D <-> (0 < D \/ 0 = D)))
11	7, 10	syl 12	. . 3 |- (D e. S -> (0 <_ D <-> (0 < D \/ 0 = D)))
12	6, 11	mpbid 170	. 2 |- (D e. S -> (0 < D \/ 0 = D))
13	5	pm3.27d 262	. . . . . . 7 |- (D e. S -> (D x. D) <_ A)
14		sqrlem9.3	. . . . . . . . 9 |- B e. RR
15		sqrlem9.4	. . . . . . . . 9 |- C e. RR
16		sqrlem9.5	. . . . . . . . 9 |- 0 < B
17		sqrlem9.6	. . . . . . . . 9 |- A < (B x. B)
18		sqrlem9.7	. . . . . . . . 9 |- C = ((B + (A / B)) / (1 + 1))
19	1, 2, 14, 15, 16, 17, 18	sqrlem11 4741	. . . . . . . 8 |- A < (C x. C)
20		axmulrcl 4069	. . . . . . . . . 10 |- ((D e. RR /\ D e. RR) -> (D x. D) e. RR)
21	20	anidms 332	. . . . . . . . 9 |- (D e. RR -> (D x. D) e. RR)
22	15, 15	remulcl 4119	. . . . . . . . . . 11 |- (C x. C) e. RR
23		lelttrt 4289	. . . . . . . . . . 11 |- (((D x. D) e. RR /\ A e. RR /\ (C x. C) e. RR) -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
24	22, 23	mp3an3 641	. . . . . . . . . 10 |- (((D x. D) e. RR /\ A e. RR) -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
25	1, 24	mpan2 519	. . . . . . . . 9 |- ((D x. D) e. RR -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
26	7, 21, 25	3syl 21	. . . . . . . 8 |- (D e. S -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
27	19, 26	mpan2i 522	. . . . . . 7 |- (D e. S -> ((D x. D) <_ A -> (D x. D) < (C x. C)))
28	13, 27	mpd 46	. . . . . 6 |- (D e. S -> (D x. D) < (C x. C))
29	28	adantr 306	. . . . 5 |- ((D e. S /\ 0 < D) -> (D x. D) < (C x. C))
30	1, 2, 14, 15, 16, 17, 18	sqrlem9 4739	. . . . . . 7 |- 0 < C
31		breq2 2066	. . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (0 < D <-> 0 < if(D e. RR, D, 0)))
32	31	anbi1d 469	. . . . . . . . . 10 |- (D = if(D e. RR, D, 0) -> ((0 < D /\ 0 < C) <-> (0 < if(D e. RR, D, 0) /\ 0 < C)))
33		breq1 2065	. . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (D < C <-> if(D e. RR, D, 0) < C))
34		opreq12 3008	. . . . . . . . . . . . 13 |- ((D = if(D e. RR, D, 0) /\ D = if(D e. RR, D, 0)) -> (D x. D) = (if(D e. RR, D, 0) x. if(D e. RR, D, 0)))
35	34	anidms 332	. . . . . . . . . . . 12 |- (D = if(D e. RR, D, 0) -> (D x. D) = (if(D e. RR, D, 0) x. if(D e. RR, D, 0)))
36	35	breq1d 2071	. . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> ((D x. D) < (C x. C) <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
37	33, 36	bibi12d 477	. . . . . . . . . 10 |- (D = if(D e. RR, D, 0) -> ((D < C <-> (D x. D) < (C x. C)) <-> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C))))
38	32, 37	imbi12d 474	. . . . . . . . 9 |- (D = if(D e. RR, D, 0) -> (((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))) <-> ((0 < if(D e. RR, D, 0) /\ 0 < C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))))
39	8	elimel 1793	. . . . . . . . . . 11 |- if(D e. RR, D, 0) e. RR
40	39, 15	lt2sq 4414	. . . . . . . . . 10 |- ((0 <_ if(D e. RR, D, 0) /\ 0 <_ C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
41	8, 39	ltle 4302	. . . . . . . . . 10 |- (0 < if(D e. RR, D, 0) -> 0 <_ if(D e. RR, D, 0))
42	8, 15	ltle 4302	. . . . . . . . . 10 |- (0 < C -> 0 <_ C)
43	40, 41, 42	syl2an 349	. . . . . . . . 9 |- ((0 < if(D e. RR, D, 0) /\ 0 < C) -> (if(D e. RR, D, 0) < C <-> (if(D e. RR, D, 0) x. if(D e. RR, D, 0)) < (C x. C)))
44	38, 43	dedth 1784	. . . . . . . 8 |- (D e. RR -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
45	7, 44	syl 12	. . . . . . 7 |- (D e. S -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
46	30, 45	mpan2i 522	. . . . . 6 |- (D e. S -> (0 < D -> (D < C <-> (D x. D) < (C x. C))))
47	46	imp 277	. . . . 5 |- ((D e. S /\ 0 < D) -> (D < C <-> (D x. D) < (C x. C)))
48	29, 47	mpbird 171	. . . 4 |- ((D e. S /\ 0 < D) -> D < C)
49	48	exp 291	. . 3 |- (D e. S -> (0 < D -> D < C))
50		breq1 2065	. . . . 5 |- (0 = D -> (0 < C <-> D < C))
51	30, 50	mpbii 168	. . . 4 |- (0 = D -> D < C)
52	51	a1i 7	. . 3 |- (D e. S -> (0 = D -> D < C))
53	49, 52	jaod 329	. 2 |- (D e. S -> ((0 < D \/ 0 = D) -> D < C))
54	12, 53	mpd 46	1 |- (D e. S -> D < C)

Syntax hints:   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  {crab 1204  ifcif 1776   class class class wbr 2054  (class class class)co 3001  RRcr 4027  0cc0 4028  1c1 4029   + caddc 4031   x. cmulc 4032   < clt 4033   / cdiv 4091   <_ cle 4092

This theorem is referenced by:  sqrlem13 4743

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277

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