Theorem sqrlem12 6685

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 groups:   x,A   x,B   x,S   x,C   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 6677	. . . . 5 |- (D e. S <-> (D e. RR /\ (0 <_ D /\ (D x. D) <_ A)))
5	4	pm3.27bi 326	. . . 4 |- (D e. S -> (0 <_ D /\ (D x. D) <_ A))
6	5	pm3.26d 321	. . 3 |- (D e. S -> 0 <_ D)
7	4	pm3.26bi 322	. . . 4 |- (D e. S -> D e. RR)
8		0re 5452	. . . . 5 |- 0 e. RR
9		leloet 5530	. . . . 5 |- ((0 e. RR /\ D e. RR) -> (0 <_ D <-> (0 < D \/ 0 = D)))
10	8, 9	mpan 697	. . . 4 |- (D e. RR -> (0 <_ D <-> (0 < D \/ 0 = D)))
11	7, 10	syl 10	. . 3 |- (D e. S -> (0 <_ D <-> (0 < D \/ 0 = D)))
12	6, 11	mpbid 195	. 2 |- (D e. S -> (0 < D \/ 0 = D))
13	5	pm3.27d 325	. . . . . . 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 6684	. . . . . . . 8 |- A < (C x. C)
20		axmulrcl 5286	. . . . . . . . . 10 |- ((D e. RR /\ D e. RR) -> (D x. D) e. RR)
21	20	anidms 436	. . . . . . . . 9 |- (D e. RR -> (D x. D) e. RR)
22	15, 15	remulcl 5347	. . . . . . . . . 10 |- (C x. C) e. RR
23		lelttrt 5535	. . . . . . . . . 10 |- (((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	1, 22, 23	mp3an23 910	. . . . . . . . 9 |- ((D x. D) e. RR -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
25	7, 21, 24	3syl 20	. . . . . . . 8 |- (D e. S -> (((D x. D) <_ A /\ A < (C x. C)) -> (D x. D) < (C x. C)))
26	19, 25	mpan2i 701	. . . . . . 7 |- (D e. S -> ((D x. D) <_ A -> (D x. D) < (C x. C)))
27	13, 26	mpd 26	. . . . . 6 |- (D e. S -> (D x. D) < (C x. C))
28	27	adantr 391	. . . . 5 |- ((D e. S /\ 0 < D) -> (D x. D) < (C x. C))
29	1, 2, 14, 15, 16, 17, 18	sqrlem9 6682	. . . . . . 7 |- 0 < C
30		breq2 2628	. . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (0 < D <-> 0 < if(D e. RR, D, 0)))
31	30	anbi1d 619	. . . . . . . . . 10 |- (D = if(D e. RR, D, 0) -> ((0 < D /\ 0 < C) <-> (0 < if(D e. RR, D, 0) /\ 0 < C)))
32		breq1 2627	. . . . . . . . . . 11 |- (D = if(D e. RR, D, 0) -> (D < C <-> if(D e. RR, D, 0) < C))
33		opreq12 3976	. . . . . . . . . . . . 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)))
34	33	anidms 436	. . . . . . . . . . . 12 |- (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	breq1d 2634	. . . . . . . . . . 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)))
36	32, 35	bibi12d 631	. . . . . . . . . 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))))
37	31, 36	imbi12d 628	. . . . . . . . 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)))))
38	8	elimel 2398	. . . . . . . . . . 11 |- if(D e. RR, D, 0) e. RR
39	38, 15	lt2msq 5883	. . . . . . . . . 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)))
40	8, 38	ltle 5592	. . . . . . . . . 10 |- (0 < if(D e. RR, D, 0) -> 0 <_ if(D e. RR, D, 0))
41	8, 15	ltle 5592	. . . . . . . . . 10 |- (0 < C -> 0 <_ C)
42	39, 40, 41	syl2an 456	. . . . . . . . 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)))
43	37, 42	dedth 2387	. . . . . . . 8 |- (D e. RR -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
44	7, 43	syl 10	. . . . . . 7 |- (D e. S -> ((0 < D /\ 0 < C) -> (D < C <-> (D x. D) < (C x. C))))
45	29, 44	mpan2i 701	. . . . . 6 |- (D e. S -> (0 < D -> (D < C <-> (D x. D) < (C x. C))))
46	45	imp 350	. . . . 5 |- ((D e. S /\ 0 < D) -> (D < C <-> (D x. D) < (C x. C)))
47	28, 46	mpbird 196	. . . 4 |- ((D e. S /\ 0 < D) -> D < C)
48	47	ex 373	. . 3 |- (D e. S -> (0 < D -> D < C))
49		breq1 2627	. . . . 5 |- (0 = D -> (0 < C <-> D < C))
50	29, 49	mpbii 193	. . . 4 |- (0 = D -> D < C)
51	50	a1i 8	. . 3 |- (D e. S -> (0 = D -> D < C))
52	48, 51	jaod 426	. 2 |- (D e. S -> ((0 < D \/ 0 = D) -> D < C))
53	12, 52	mpd 26	1 |- (D e. S -> D < C)

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960  {crab 1651  ifcif 2365   class class class wbr 2624  (class class class)co 3969  RRcr 5245  0cc0 5246  1c1 5247   + caddc 5249   x. cmul 5251   / cdiv 5306   <_ cle 5307   < clt 5498

This theorem is referenced by:  sqrlem13 6686

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

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-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

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