Theorem lt2sq 4414

Description: Two nonnegative numbers compare the same as their squares.

Hypotheses

Ref	Expression
ltrec.1	|- A e. RR
ltrec.2	|- B e. RR

Assertion

Ref	Expression
lt2sq	|- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Proof of Theorem lt2sq

Step	Hyp	Ref	Expression
1		ltrec.1	. . . . . . . 8 |- A e. RR
2		ltrec.2	. . . . . . . 8 |- B e. RR
3	1, 2, 1	ltmul2 4395	. . . . . . 7 |- (0 < A -> (A < B <-> (A x. A) < (A x. B)))
4	1, 2, 2	ltmul1 4394	. . . . . . 7 |- (0 < B -> (A < B <-> (A x. B) < (B x. B)))
5	3, 4	bi2anan9 478	. . . . . 6 |- ((0 < A /\ 0 < B) -> ((A < B /\ A < B) <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
6		anidm 331	. . . . . 6 |- ((A < B /\ A < B) <-> A < B)
7	5, 6	syl5bbr 412	. . . . 5 |- ((0 < A /\ 0 < B) -> (A < B <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
8	1, 1	remulcl 4119	. . . . . 6 |- (A x. A) e. RR
9	1, 2	remulcl 4119	. . . . . 6 |- (A x. B) e. RR
10	2, 2	remulcl 4119	. . . . . 6 |- (B x. B) e. RR
11	8, 9, 10	lttr 4307	. . . . 5 |- (((A x. A) < (A x. B) /\ (A x. B) < (B x. B)) -> (A x. A) < (B x. B))
12	7, 11	syl6bi 187	. . . 4 |- ((0 < A /\ 0 < B) -> (A < B -> (A x. A) < (B x. B)))
13	2, 1, 2	lemul2 4396	. . . . . . . . 9 |- (0 < B -> (B <_ A <-> (B x. B) <_ (B x. A)))
14	2, 1, 1	lemul1 4397	. . . . . . . . 9 |- (0 < A -> (B <_ A <-> (B x. A) <_ (A x. A)))
15	13, 14	bi2anan9r 479	. . . . . . . 8 |- ((0 < A /\ 0 < B) -> ((B <_ A /\ B <_ A) <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
16		anidm 331	. . . . . . . 8 |- ((B <_ A /\ B <_ A) <-> B <_ A)
17	15, 16	syl5bbr 412	. . . . . . 7 |- ((0 < A /\ 0 < B) -> (B <_ A <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
18	2, 1	remulcl 4119	. . . . . . . 8 |- (B x. A) e. RR
19	10, 18, 8	letr 4310	. . . . . . 7 |- (((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A)) -> (B x. B) <_ (A x. A))
20	17, 19	syl6bi 187	. . . . . 6 |- ((0 < A /\ 0 < B) -> (B <_ A -> (B x. B) <_ (A x. A)))
21	2, 1	lelt 4301	. . . . . 6 |- (B <_ A <-> -. A < B)
22	10, 8	lelt 4301	. . . . . 6 |- ((B x. B) <_ (A x. A) <-> -. (A x. A) < (B x. B))
23	20, 21, 22	3imtr3g 425	. . . . 5 |- ((0 < A /\ 0 < B) -> (-. A < B -> -. (A x. A) < (B x. B)))
24	23	a3d 70	. . . 4 |- ((0 < A /\ 0 < B) -> ((A x. A) < (B x. B) -> A < B))
25	12, 24	impbid 397	. . 3 |- ((0 < A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
26		breq1 2065	. . . . 5 |- (0 = A -> (0 < B <-> A < B))
27	26	adantr 306	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> A < B))
28		ax0re 4063	. . . . . 6 |- 0 e. RR
29	28, 2, 2	ltmul2 4395	. . . . 5 |- (0 < B -> (0 < B <-> (B x. 0) < (B x. B)))
30		opreq2 3007	. . . . . . 7 |- (0 = A -> (A x. 0) = (A x. A))
31	30	breq1d 2071	. . . . . 6 |- (0 = A -> ((A x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
32	2	recn 4098	. . . . . . . . 9 |- B e. CC
33	32	mulzer1 4185	. . . . . . . 8 |- (B x. 0) = 0
34	1	recn 4098	. . . . . . . . 9 |- A e. CC
35	34	mulzer1 4185	. . . . . . . 8 |- (A x. 0) = 0
36	33, 35	eqtr4 1122	. . . . . . 7 |- (B x. 0) = (A x. 0)
37	36	breq1i 2068	. . . . . 6 |- ((B x. 0) < (B x. B) <-> (A x. 0) < (B x. B))
38	31, 37	syl5bb 410	. . . . 5 |- (0 = A -> ((B x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
39	29, 38	sylan9bbr 419	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> (A x. A) < (B x. B)))
40	27, 39	bitr3d 408	. . 3 |- ((0 = A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
41		breq1 2065	. . . . . . 7 |- (0 = B -> (0 <_ A <-> B <_ A))
42	41	adantl 305	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> B <_ A))
43	28, 1, 1	lemul2 4396	. . . . . . 7 |- (0 < A -> (0 <_ A <-> (A x. 0) <_ (A x. A)))
44		opreq2 3007	. . . . . . . . 9 |- (0 = B -> (B x. 0) = (B x. B))
45	44	breq1d 2071	. . . . . . . 8 |- (0 = B -> ((B x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
46	35, 33	eqtr4 1122	. . . . . . . . 9 |- (A x. 0) = (B x. 0)
47	46	breq1i 2068	. . . . . . . 8 |- ((A x. 0) <_ (A x. A) <-> (B x. 0) <_ (A x. A))
48	45, 47	syl5bb 410	. . . . . . 7 |- (0 = B -> ((A x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
49	43, 48	sylan9bb 418	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> (B x. B) <_ (A x. A)))
50	42, 49	bitr3d 408	. . . . 5 |- ((0 < A /\ 0 = B) -> (B <_ A <-> (B x. B) <_ (A x. A)))
51	50, 21, 22	3bitr3g 427	. . . 4 |- ((0 < A /\ 0 = B) -> (-. A < B <-> -. (A x. A) < (B x. B)))
52	51	bicon4d 402	. . 3 |- ((0 < A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
53		pm5.21 502	. . . 4 |- ((-. A < B /\ -. (A x. A) < (B x. B)) -> (A < B <-> (A x. A) < (B x. B)))
54	2	ltnr 4338	. . . . 5 |- -. B < B
55		breq1 2065	. . . . . . 7 |- (0 = B -> (0 < B <-> B < B))
56	55	bicomd 399	. . . . . 6 |- (0 = B -> (B < B <-> 0 < B))
57	56, 26	sylan9bbr 419	. . . . 5 |- ((0 = A /\ 0 = B) -> (B < B <-> A < B))
58	54, 57	mtbii 538	. . . 4 |- ((0 = A /\ 0 = B) -> -. A < B)
59	10	ltnr 4338	. . . . 5 |- -. (B x. B) < (B x. B)
60	44	breq1d 2071	. . . . . . 7 |- (0 = B -> ((B x. 0) < (B x. B) <-> (B x. B) < (B x. B)))
61	60	bicomd 399	. . . . . 6 |- (0 = B -> ((B x. B) < (B x. B) <-> (B x. 0) < (B x. B)))
62	61, 38	sylan9bbr 419	. . . . 5 |- ((0 = A /\ 0 = B) -> ((B x. B) < (B x. B) <-> (A x. A) < (B x. B)))
63	59, 62	mtbii 538	. . . 4 |- ((0 = A /\ 0 = B) -> -. (A x. A) < (B x. B))
64	53, 58, 63	sylanc 361	. . 3 |- ((0 = A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
65	25, 40, 52, 64	ccase 562	. 2 |- (((0 < A \/ 0 = A) /\ (0 < B \/ 0 = B)) -> (A < B <-> (A x. A) < (B x. B)))
66	28, 1	leloe 4298	. 2 |- (0 <_ A <-> (0 < A \/ 0 = A))
67	28, 2	leloe 4298	. 2 |- (0 <_ B <-> (0 < B \/ 0 = B))
68	65, 66, 67	syl2anb 350	1 |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   class class class wbr 2054  (class class class)co 3001  RRcr 4027  0cc0 4028   x. cmulc 4032   < clt 4033   <_ cle 4092

This theorem is referenced by:  le2sq 4415  sq11 4416  lt2sqe 4700  sqrlem6 4736  sqrlem12 4742

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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