Theorem resq01 11027

Description: If a real number equals its square, it must be 0 or 1. (Contributed by Jim Kingdon, 2-Jun-2026.)

Assertion

Ref	Expression
resq01	|- ( A e. RR -> ( ( A ^ 2 ) = A <-> ( A = 0 \/ A = 1 ) ) )

Proof of Theorem resq01

Step	Hyp	Ref	Expression
1		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A e. RR )
2	1	recnd 8307	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A e. CC )
3		sqval 10966	. . . . . . . . 9 |- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) )
4	2, 3	syl 14	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A ^ 2 ) = ( A x. A ) )
5		simpr 110	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A ^ 2 ) = A )
6	4, 5	eqtr3d 2269	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A x. A ) = A )
7		simplr 529	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> 0 < A )
8	1, 7	gt0ap0d 8908	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A # 0 )
9	2, 2, 2, 8	divmulapd 9091	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( ( A / A ) = A <-> ( A x. A ) = A ) )
10	6, 9	mpbird 167	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A / A ) = A )
11	2, 8	dividapd 9065	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A / A ) = 1 )
12	10, 11	eqtr3d 2269	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A = 1 )
13	12	olcd 742	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A = 0 \/ A = 1 ) )
14	13	ex 115	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( ( A ^ 2 ) = A -> ( A = 0 \/ A = 1 ) ) )
15		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A e. RR )
16	15	recnd 8307	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A e. CC )
17	16, 16	muls1d 8696	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. ( A - 1 ) ) = ( ( A x. A ) - A ) )
18	16, 16	mulcld 8299	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. A ) e. CC )
19	16, 3	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A ^ 2 ) = ( A x. A ) )
20		simpr 110	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A ^ 2 ) = A )
21	19, 20	eqtr3d 2269	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. A ) = A )
22	18, 21	subeq0bd 8657	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( ( A x. A ) - A ) = 0 )
23	17, 22	eqtr2d 2268	. . . . . . 7 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 0 = ( A x. ( A - 1 ) ) )
24		0cnd 8272	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 0 e. CC )
25		1red 8294	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 1 e. RR )
26	15, 25	resubcld 8659	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) e. RR )
27	26	recnd 8307	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) e. CC )
28		simplr 529	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A < 1 )
29	15, 25	sublt0d 8849	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( ( A - 1 ) < 0 <-> A < 1 ) )
30	28, 29	mpbird 167	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) < 0 )
31	26, 30	lt0ap0d 8928	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) # 0 )
32	24, 16, 27, 31	divmulap3d 9104	. . . . . . 7 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( ( 0 / ( A - 1 ) ) = A <-> 0 = ( A x. ( A - 1 ) ) ) )
33	23, 32	mpbird 167	. . . . . 6 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( 0 / ( A - 1 ) ) = A )
34	27, 31	div0apd 9066	. . . . . 6 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( 0 / ( A - 1 ) ) = 0 )
35	33, 34	eqtr3d 2269	. . . . 5 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A = 0 )
36	35	orcd 741	. . . 4 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A = 0 \/ A = 1 ) )
37	36	ex 115	. . 3 |- ( ( A e. RR /\ A < 1 ) -> ( ( A ^ 2 ) = A -> ( A = 0 \/ A = 1 ) ) )
38		0lt1 8405	. . . 4 |- 0 < 1
39		0re 8279	. . . . 5 |- 0 e. RR
40		1re 8278	. . . . 5 |- 1 e. RR
41		axltwlin 8346	. . . . 5 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( 0 < 1 -> ( 0 < A \/ A < 1 ) ) )
42	39, 40, 41	mp3an12 1364	. . . 4 |- ( A e. RR -> ( 0 < 1 -> ( 0 < A \/ A < 1 ) ) )
43	38, 42	mpi 15	. . 3 |- ( A e. RR -> ( 0 < A \/ A < 1 ) )
44	14, 37, 43	mpjaodan 806	. 2 |- ( A e. RR -> ( ( A ^ 2 ) = A -> ( A = 0 \/ A = 1 ) ) )
45		sq0 10999	. . . 4 |- ( 0 ^ 2 ) = 0
46		oveq1 6059	. . . 4 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
47		id 19	. . . 4 |- ( A = 0 -> A = 0 )
48	45, 46, 47	3eqtr4a 2293	. . 3 |- ( A = 0 -> ( A ^ 2 ) = A )
49		sq1 11002	. . . 4 |- ( 1 ^ 2 ) = 1
50		oveq1 6059	. . . 4 |- ( A = 1 -> ( A ^ 2 ) = ( 1 ^ 2 ) )
51		id 19	. . . 4 |- ( A = 1 -> A = 1 )
52	49, 50, 51	3eqtr4a 2293	. . 3 |- ( A = 1 -> ( A ^ 2 ) = A )
53	48, 52	jaoi 724	. 2 |- ( ( A = 0 \/ A = 1 ) -> ( A ^ 2 ) = A )
54	44, 53	impbid1 142	1 |- ( A e. RR -> ( ( A ^ 2 ) = A <-> ( A = 0 \/ A = 1 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8130   RRcr 8131   0cc0 8132   1c1 8133   x. cmul 8137   < clt 8313   - cmin 8449   / cdiv 8951   2c2 9293   ^cexp 10907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-n0 9502  df-z 9583  df-uz 9860  df-seqfrec 10817  df-exp 10908

This theorem is referenced by: (None)