Theorem lt2msqi 7064

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
lt2msqi	|- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Proof of Theorem lt2msqi

Step	Hyp	Ref	Expression
1		ltrec.1	. . . . . . . 8 |- A e. RR
2		ltrec.2	. . . . . . . 8 |- B e. RR
3	1, 2, 1	ltmul2i 7015	. . . . . . 7 |- (0 < A -> (A < B <-> (A x. A) < (A x. B)))
4	1, 2, 2	ltmul1i 7000	. . . . . . 7 |- (0 < B -> (A < B <-> (A x. B) < (B x. B)))
5	3, 4	bi2anan9 694	. . . . . 6 |- ((0 < A /\ 0 < B) -> ((A < B /\ A < B) <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
6		anidm 478	. . . . . 6 |- ((A < B /\ A < B) <-> A < B)
7	5, 6	syl5bbr 593	. . . . 5 |- ((0 < A /\ 0 < B) -> (A < B <-> ((A x. A) < (A x. B) /\ (A x. B) < (B x. B))))
8	1, 1	remulcli 6488	. . . . . 6 |- (A x. A) e. RR
9	1, 2	remulcli 6488	. . . . . 6 |- (A x. B) e. RR
10	2, 2	remulcli 6488	. . . . . 6 |- (B x. B) e. RR
11	8, 9, 10	lttri 6760	. . . . 5 |- (((A x. A) < (A x. B) /\ (A x. B) < (B x. B)) -> (A x. A) < (B x. B))
12	7, 11	syl6bi 231	. . . 4 |- ((0 < A /\ 0 < B) -> (A < B -> (A x. A) < (B x. B)))
13	2, 1, 2	lemul2i 7018	. . . . . . . 8 |- (0 < B -> (B <_ A <-> (B x. B) <_ (B x. A)))
14	2, 1, 1	lemul1i 7017	. . . . . . . 8 |- (0 < A -> (B <_ A <-> (B x. A) <_ (A x. A)))
15	13, 14	bi2anan9r 695	. . . . . . 7 |- ((0 < A /\ 0 < B) -> ((B <_ A /\ B <_ A) <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
16		anidm 478	. . . . . . 7 |- ((B <_ A /\ B <_ A) <-> B <_ A)
17	15, 16	syl5bbr 593	. . . . . 6 |- ((0 < A /\ 0 < B) -> (B <_ A <-> ((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A))))
18	2, 1	remulcli 6488	. . . . . . 7 |- (B x. A) e. RR
19	10, 18, 8	letri 6763	. . . . . 6 |- (((B x. B) <_ (B x. A) /\ (B x. A) <_ (A x. A)) -> (B x. B) <_ (A x. A))
20	17, 19	syl6bi 231	. . . . 5 |- ((0 < A /\ 0 < B) -> (B <_ A -> (B x. B) <_ (A x. A)))
21	2, 1	lenlti 6753	. . . . 5 |- (B <_ A <-> -. A < B)
22	10, 8	lenlti 6753	. . . . 5 |- ((B x. B) <_ (A x. A) <-> -. (A x. A) < (B x. B))
23	20, 21, 22	3imtr3g 611	. . . 4 |- ((0 < A /\ 0 < B) -> (-. A < B -> -. (A x. A) < (B x. B)))
24	12, 23	impcon4bid 578	. . 3 |- ((0 < A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
25		breq1 3341	. . . . 5 |- (0 = A -> (0 < B <-> A < B))
26	25	adantr 425	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> A < B))
27		0re 6603	. . . . . 6 |- 0 e. RR
28	27, 2, 2	ltmul2i 7015	. . . . 5 |- (0 < B -> (0 < B <-> (B x. 0) < (B x. B)))
29		opreq2 4890	. . . . . . 7 |- (0 = A -> (A x. 0) = (A x. A))
30	29	breq1d 3348	. . . . . 6 |- (0 = A -> ((A x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
31	2	recni 6467	. . . . . . . . 9 |- B e. CC
32	31	mul01i 6594	. . . . . . . 8 |- (B x. 0) = 0
33	1	recni 6467	. . . . . . . . 9 |- A e. CC
34	33	mul01i 6594	. . . . . . . 8 |- (A x. 0) = 0
35	32, 34	eqtr4i 1911	. . . . . . 7 |- (B x. 0) = (A x. 0)
36	35	breq1i 3345	. . . . . 6 |- ((B x. 0) < (B x. B) <-> (A x. 0) < (B x. B))
37	30, 36	syl5bb 591	. . . . 5 |- (0 = A -> ((B x. 0) < (B x. B) <-> (A x. A) < (B x. B)))
38	28, 37	sylan9bbr 600	. . . 4 |- ((0 = A /\ 0 < B) -> (0 < B <-> (A x. A) < (B x. B)))
39	26, 38	bitr3d 589	. . 3 |- ((0 = A /\ 0 < B) -> (A < B <-> (A x. A) < (B x. B)))
40		breq1 3341	. . . . . . 7 |- (0 = B -> (0 <_ A <-> B <_ A))
41	40	adantl 424	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> B <_ A))
42	27, 1, 1	lemul2i 7018	. . . . . . 7 |- (0 < A -> (0 <_ A <-> (A x. 0) <_ (A x. A)))
43		opreq2 4890	. . . . . . . . 9 |- (0 = B -> (B x. 0) = (B x. B))
44	43	breq1d 3348	. . . . . . . 8 |- (0 = B -> ((B x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
45	34, 32	eqtr4i 1911	. . . . . . . . 9 |- (A x. 0) = (B x. 0)
46	45	breq1i 3345	. . . . . . . 8 |- ((A x. 0) <_ (A x. A) <-> (B x. 0) <_ (A x. A))
47	44, 46	syl5bb 591	. . . . . . 7 |- (0 = B -> ((A x. 0) <_ (A x. A) <-> (B x. B) <_ (A x. A)))
48	42, 47	sylan9bb 599	. . . . . 6 |- ((0 < A /\ 0 = B) -> (0 <_ A <-> (B x. B) <_ (A x. A)))
49	41, 48	bitr3d 589	. . . . 5 |- ((0 < A /\ 0 = B) -> (B <_ A <-> (B x. B) <_ (A x. A)))
50	49, 21, 22	3bitr3g 613	. . . 4 |- ((0 < A /\ 0 = B) -> (-. A < B <-> -. (A x. A) < (B x. B)))
51	50	con4bid 583	. . 3 |- ((0 < A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
52	2	ltnri 6789	. . . . 5 |- -. B < B
53		breq1 3341	. . . . . . 7 |- (0 = B -> (0 < B <-> B < B))
54	53	bicomd 580	. . . . . 6 |- (0 = B -> (B < B <-> 0 < B))
55	54, 25	sylan9bbr 600	. . . . 5 |- ((0 = A /\ 0 = B) -> (B < B <-> A < B))
56	52, 55	mtbii 784	. . . 4 |- ((0 = A /\ 0 = B) -> -. A < B)
57	10	ltnri 6789	. . . . 5 |- -. (B x. B) < (B x. B)
58	43	breq1d 3348	. . . . . . 7 |- (0 = B -> ((B x. 0) < (B x. B) <-> (B x. B) < (B x. B)))
59	58	bicomd 580	. . . . . 6 |- (0 = B -> ((B x. B) < (B x. B) <-> (B x. 0) < (B x. B)))
60	59, 37	sylan9bbr 600	. . . . 5 |- ((0 = A /\ 0 = B) -> ((B x. B) < (B x. B) <-> (A x. A) < (B x. B)))
61	57, 60	mtbii 784	. . . 4 |- ((0 = A /\ 0 = B) -> -. (A x. A) < (B x. B))
62		pm5.21 741	. . . 4 |- ((-. A < B /\ -. (A x. A) < (B x. B)) -> (A < B <-> (A x. A) < (B x. B)))
63	56, 61, 62	syl11anc 524	. . 3 |- ((0 = A /\ 0 = B) -> (A < B <-> (A x. A) < (B x. B)))
64	24, 39, 51, 63	ccase 829	. 2 |- (((0 < A \/ 0 = A) /\ (0 < B \/ 0 = B)) -> (A < B <-> (A x. A) < (B x. B)))
65	27, 1	leloei 6750	. 2 |- (0 <_ A <-> (0 < A \/ 0 = A))
66	27, 2	leloei 6750	. 2 |- (0 <_ B <-> (0 < B \/ 0 = B))
67	64, 65, 66	syl2anb 504	1 |- ((0 <_ A /\ 0 <_ B) -> (A < B <-> (A x. A) < (B x. B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386   x. cmul 6391   <_ cle 6448   < clt 6653

This theorem is referenced by:  le2msqi 7065  msq11i 7066  lt2msq 7069  lt2sqi 7869  sqrlem6 7928  sqrlem12 7934

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658

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