Theorem sqrlearg 39780

Description: The square compared with its argument. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
sqrlearg.1	|- ( ph -> A e. RR )

Assertion

Ref	Expression
sqrlearg	|- ( ph -> ( ( A ^ 2 ) <_ A <-> A e. ( 0 [,] 1 ) ) )

Proof of Theorem sqrlearg

Step	Hyp	Ref	Expression
1		0re 10040	. . . . 5 |- 0 e. RR
2	1	a1i 11	. . . 4 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 e. RR )
3		simpr 477	. . . . . . . . 9 |- ( ( ph /\ -. A <_ 1 ) -> -. A <_ 1 )
4		1red 10055	. . . . . . . . . 10 |- ( ( ph /\ -. A <_ 1 ) -> 1 e. RR )
5		sqrlearg.1	. . . . . . . . . . 11 |- ( ph -> A e. RR )
6	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ -. A <_ 1 ) -> A e. RR )
7	4, 6	ltnled 10184	. . . . . . . . 9 |- ( ( ph /\ -. A <_ 1 ) -> ( 1 < A <-> -. A <_ 1 ) )
8	3, 7	mpbird 247	. . . . . . . 8 |- ( ( ph /\ -. A <_ 1 ) -> 1 < A )
9		1red 10055	. . . . . . . . . 10 |- ( ( ph /\ 1 < A ) -> 1 e. RR )
10	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ 1 < A ) -> A e. RR )
11	1	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> 0 e. RR )
12		0lt1 10550	. . . . . . . . . . . . 13 |- 0 < 1
13	12	a1i 11	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> 0 < 1 )
14		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ 1 < A ) -> 1 < A )
15	11, 9, 10, 13, 14	lttrd 10198	. . . . . . . . . . 11 |- ( ( ph /\ 1 < A ) -> 0 < A )
16	10, 15	elrpd 11869	. . . . . . . . . 10 |- ( ( ph /\ 1 < A ) -> A e. RR+ )
17	9, 10, 16, 14	ltmul2dd 11928	. . . . . . . . 9 |- ( ( ph /\ 1 < A ) -> ( A x. 1 ) < ( A x. A ) )
18	5	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
19	18	mulid1d 10057	. . . . . . . . . . 11 |- ( ph -> ( A x. 1 ) = A )
20	19	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ 1 < A ) -> ( A x. 1 ) = A )
21	18	sqvald 13005	. . . . . . . . . . . 12 |- ( ph -> ( A ^ 2 ) = ( A x. A ) )
22	21	eqcomd 2628	. . . . . . . . . . 11 |- ( ph -> ( A x. A ) = ( A ^ 2 ) )
23	22	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ 1 < A ) -> ( A x. A ) = ( A ^ 2 ) )
24	20, 23	breq12d 4666	. . . . . . . . 9 |- ( ( ph /\ 1 < A ) -> ( ( A x. 1 ) < ( A x. A ) <-> A < ( A ^ 2 ) ) )
25	17, 24	mpbid 222	. . . . . . . 8 |- ( ( ph /\ 1 < A ) -> A < ( A ^ 2 ) )
26	8, 25	syldan 487	. . . . . . 7 |- ( ( ph /\ -. A <_ 1 ) -> A < ( A ^ 2 ) )
27	26	adantlr 751	. . . . . 6 |- ( ( ( ph /\ ( A ^ 2 ) <_ A ) /\ -. A <_ 1 ) -> A < ( A ^ 2 ) )
28		simpr 477	. . . . . . . 8 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( A ^ 2 ) <_ A )
29	5	resqcld 13035	. . . . . . . . . 10 |- ( ph -> ( A ^ 2 ) e. RR )
30	29	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( A ^ 2 ) e. RR )
31	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A e. RR )
32	30, 31	lenltd 10183	. . . . . . . 8 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> ( ( A ^ 2 ) <_ A <-> -. A < ( A ^ 2 ) ) )
33	28, 32	mpbid 222	. . . . . . 7 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> -. A < ( A ^ 2 ) )
34	33	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( A ^ 2 ) <_ A ) /\ -. A <_ 1 ) -> -. A < ( A ^ 2 ) )
35	27, 34	condan 835	. . . . 5 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A <_ 1 )
36		1red 10055	. . . . 5 |- ( ( ph /\ A <_ 1 ) -> 1 e. RR )
37	35, 36	syldan 487	. . . 4 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 1 e. RR )
38	31	sqge0d 13036	. . . . 5 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 <_ ( A ^ 2 ) )
39	2, 30, 31, 38, 28	letrd 10194	. . . 4 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> 0 <_ A )
40	2, 37, 31, 39, 35	eliccd 39726	. . 3 |- ( ( ph /\ ( A ^ 2 ) <_ A ) -> A e. ( 0 [,] 1 ) )
41	40	ex 450	. 2 |- ( ph -> ( ( A ^ 2 ) <_ A -> A e. ( 0 [,] 1 ) ) )
42		unitssre 12319	. . . . . . 7 |- ( 0 [,] 1 ) C_ RR
43	42	sseli 3599	. . . . . 6 |- ( A e. ( 0 [,] 1 ) -> A e. RR )
44		1red 10055	. . . . . 6 |- ( A e. ( 0 [,] 1 ) -> 1 e. RR )
45		0xr 10086	. . . . . . . 8 |- 0 e. RR*
46	45	a1i 11	. . . . . . 7 |- ( A e. ( 0 [,] 1 ) -> 0 e. RR* )
47	44	rexrd 10089	. . . . . . 7 |- ( A e. ( 0 [,] 1 ) -> 1 e. RR* )
48		id 22	. . . . . . 7 |- ( A e. ( 0 [,] 1 ) -> A e. ( 0 [,] 1 ) )
49	46, 47, 48	iccgelbd 39770	. . . . . 6 |- ( A e. ( 0 [,] 1 ) -> 0 <_ A )
50	46, 47, 48	iccleubd 39775	. . . . . 6 |- ( A e. ( 0 [,] 1 ) -> A <_ 1 )
51	43, 44, 43, 49, 50	lemul2ad 10964	. . . . 5 |- ( A e. ( 0 [,] 1 ) -> ( A x. A ) <_ ( A x. 1 ) )
52	51	adantl 482	. . . 4 |- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. A ) <_ ( A x. 1 ) )
53	22	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. A ) = ( A ^ 2 ) )
54	19	adantr 481	. . . . 5 |- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A x. 1 ) = A )
55	53, 54	breq12d 4666	. . . 4 |- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( ( A x. A ) <_ ( A x. 1 ) <-> ( A ^ 2 ) <_ A ) )
56	52, 55	mpbid 222	. . 3 |- ( ( ph /\ A e. ( 0 [,] 1 ) ) -> ( A ^ 2 ) <_ A )
57	56	ex 450	. 2 |- ( ph -> ( A e. ( 0 [,] 1 ) -> ( A ^ 2 ) <_ A ) )
58	41, 57	impbid 202	1 |- ( ph -> ( ( A ^ 2 ) <_ A <-> A e. ( 0 [,] 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   2c2 11070   [,]cicc 12178   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-icc 12182  df-seq 12802  df-exp 12861

This theorem is referenced by:  smfmullem1  40998