Theorem lt2msq 5829

Description: The square function on nonnegative reals is strictly monotonic.

Hypotheses

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

Assertion

Ref	Expression
lt2msq	|- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Proof of Theorem lt2msq

Step	Hyp	Ref	Expression
1		ltrec.1	. . . . . . . 8 |- A e. RR
2		ltrec.2	. . . . . . . 8 |- B e. RR
3	1, 2, 1	ltmul2 5790	. . . . . . 7 |- (0 < A -> (A < B <-> (A x. A) < (A x. B)))
4	1, 2, 2	ltmul1 5778	. . . . . . 7 |- (0 < B -> (A < B <-> (A x. B) < (B x. B)))
5	3, 4	bi2anan9 630	. . . . . 6 |- ((0 < A /\ 0 < B) -> ((A < B /\ A < B) <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
6		anidm 432	. . . . . 6 |- ((A < B /\ A < B) <-> A < B)
7	5, 6	syl5bbr 532	. . . . 5 |- ((0 < A /\ 0 < B) -> (A < B <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
8	1, 1	remulcl 5307	. . . . . 6 |- (A x. A) e. RR
9	1, 2	remulcl 5307	. . . . . 6 |- (A x. B) e. RR
10	2, 2	remulcl 5307	. . . . . 6 |- (B x. B) e. RR
11	8, 9, 10	lttr 5559	. . . . 5 |- (((A x. A) < (A x. B) /\ (A x. B) < (B x. B)) -> (A x. A) < (B x. B))
12	7, 11	syl6bi 214	. . . 4 |- ((0 < A /\ 0 < B) -> (A < B -> (A x. A) < (B x. B)))
13	2, 1, 2	lemul2 5792	. . . . . . . . 9 |- (0 < B -> (B <_ A <-> (B x. B) <_ (B x. A)))
14	2, 1, 1	lemul1 5791	. . . . . . . . 9 |- (0 < A -> (B <_ A <-> (B x. A) <_ (A x. A)))
15	13, 14	bi2anan9r 631	. . . . . . . 8 |- ((0 < A /\ 0 < B) -> ((B <_ A /\ B <_ A) <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
16		anidm 432	. . . . . . . 8 |- ((B <_ A /\ B <_ A) <-> B <_ A)
17	15, 16	syl5bbr 532	. . . . . . 7 |- ((0 < A /\ 0 < B) -> (B <_ A <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
18	2, 1	remulcl 5307	. . . . . . . 8 |- (B x. A) e. RR
19	10, 18, 8	letr 5562	. . . . . . 7 |- (((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A)) -> (B x. B) <_ (A x. A))
20	17, 19	syl6bi 214	. . . . . 6 |- ((0 < A /\ 0 < B) -> (B <_ A -> (B x. B) <_ (A x. A)))
21	2, 1	lenlt 5551	. . . . . 6 |- (B <_ A <-> -. A < B)
22	10, 8	lenlt 5551	. . . . . 6 |- ((B x. B) <_ (A x. A) <-> -. (A x. A) < (B x. B))
23	20, 21, 22	3imtr3g 550	. . . . 5 |- ((0 < A /\ 0 < B) -> (-. A < B -> -. (A x. A) < (B x. B)))
24	23	a3d 75	. . . 4 |- ((0 < A /\ 0 < B) -> ((A x. A) < (B x. B) -> A < B))
25	12, 24	impbid 514	. . 3 |- ((0 < A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
26		breq1 2612	. . . . 5 |- (0 = A -> (0 < B <-> A < B))
27	26	adantr 389	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> A < B))
28		0re 5412	. . . . . 6 |- 0 e. RR
29	28, 2, 2	ltmul2 5790	. . . . 5 |- (0 < B -> (0 < B <-> (B x. 0) < (B x. B)))
30		opreq2 3954	. . . . . . 7 |- (0 = A -> (A x. 0) = (A x. A))
31	30	breq1d 2619	. . . . . 6 |- (0 = A -> ((A x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
32	2	recn 5286	. . . . . . . . 9 |- B e. CC
33	32	mul01 5403	. . . . . . . 8 |- (B x. 0) = 0
34	1	recn 5286	. . . . . . . . 9 |- A e. CC
35	34	mul01 5403	. . . . . . . 8 |- (A x. 0) = 0
36	33, 35	eqtr4 1490	. . . . . . 7 |- (B x. 0) = (A x. 0)
37	36	breq1i 2616	. . . . . 6 |- ((B x. 0) < (B x. B) <-> (A x. 0) < (B x. B))
38	31, 37	syl5bb 530	. . . . 5 |- (0 = A -> ((B x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
39	29, 38	sylan9bbr 539	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> (A x. A) < (B x. B)))
40	27, 39	bitr3d 528	. . 3 |- ((0 = A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
41		breq1 2612	. . . . . . 7 |- (0 = B -> (0 <_ A <-> B <_ A))
42	41	adantl 388	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> B <_ A))
43	28, 1, 1	lemul2 5792	. . . . . . 7 |- (0 < A -> (0 <_ A <-> (A x. 0) <_ (A x. A)))
44		opreq2 3954	. . . . . . . . 9 |- (0 = B -> (B x. 0) = (B x. B))
45	44	breq1d 2619	. . . . . . . 8 |- (0 = B -> ((B x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
46	35, 33	eqtr4 1490	. . . . . . . . 9 |- (A x. 0) = (B x. 0)
47	46	breq1i 2616	. . . . . . . 8 |- ((A x. 0) <_ (A x. A) <-> (B x. 0) <_ (A x. A))
48	45, 47	syl5bb 530	. . . . . . 7 |- (0 = B -> ((A x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
49	43, 48	sylan9bb 538	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> (B x. B) <_ (A x. A)))
50	42, 49	bitr3d 528	. . . . 5 |- ((0 < A /\ 0 = B) -> (B <_ A <-> (B x. B) <_ (A x. A)))
51	50, 21, 22	3bitr3g 552	. . . 4 |- ((0 < A /\ 0 = B) -> (-. A < B <-> -. (A x. A) < (B x. B)))
52	51	con4bid 522	. . 3 |- ((0 < A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
53		pm5.21 675	. . . 4 |- ((-. A < B /\ -. (A x. A) < (B x. B)) -> (A < B <-> (A x. A) < (B x. B)))
54	2	ltnr 5583	. . . . 5 |- -. B < B
55		breq1 2612	. . . . . . 7 |- (0 = B -> (0 < B <-> B < B))
56	55	bicomd 519	. . . . . 6 |- (0 = B -> (B < B <-> 0 < B))
57	56, 26	sylan9bbr 539	. . . . 5 |- ((0 = A /\ 0 = B) -> (B < B <-> A < B))
58	54, 57	mtbii 714	. . . 4 |- ((0 = A /\ 0 = B) -> -. A < B)
59	10	ltnr 5583	. . . . 5 |- -. (B x. B) < (B x. B)
60	44	breq1d 2619	. . . . . . 7 |- (0 = B -> ((B x. 0) < (B x. B) <-> (B x. B) < (B x. B)))
61	60	bicomd 519	. . . . . 6 |- (0 = B -> ((B x. B) < (B x. B) <-> (B x. 0) < (B x. B)))
62	61, 38	sylan9bbr 539	. . . . 5 |- ((0 = A /\ 0 = B) -> ((B x. B) < (B x. B) <-> (A x. A) < (B x. B)))
63	59, 62	mtbii 714	. . . 4 |- ((0 = A /\ 0 = B) -> -. (A x. A) < (B x. B))
64	53, 58, 63	sylanc 471	. . 3 |- ((0 = A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
65	25, 40, 52, 64	ccase 753	. 2 |- (((0 < A \/ 0 = A) /\ (0 < B \/ 0 = B)) -> (A < B <-> (A x. A) < (B x. B)))
66	28, 1	leloe 5548	. 2 |- (0 <_ A <-> (0 < A \/ 0 = A))
67	28, 2	leloe 5548	. 2 |- (0 <_ B <-> (0 < B \/ 0 = B))
68	65, 66, 67	syl2anb 455	1 |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   class class class wbr 2609  (class class class)co 3948  RRcr 5205  0cc0 5206   x. cmul 5211   <_ cle 5267   < clt 5458

This theorem is referenced by:  le2msq 5830  msq11 5831  lt2msqt 5834  lt2sq 6555  sqrlem6 6608  sqrlem12 6614

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463

Copyright terms: Public domain