Theorem resq01 11047

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 8318	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A e. CC )
3		sqval 10986	. . . . . . . . 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 8921	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A # 0 )
9	2, 2, 2, 8	divmulapd 9106	. . . . . . 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 9080	. . . . . 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 8318	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A e. CC )
17	16, 16	muls1d 8709	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. ( A - 1 ) ) = ( ( A x. A ) - A ) )
18	16, 16	mulcld 8310	. . . . . . . . 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 8670	. . . . . . . 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 8283	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 0 e. CC )
25		1red 8305	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 1 e. RR )
26	15, 25	resubcld 8672	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) e. RR )
27	26	recnd 8318	. . . . . . . 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 8862	. . . . . . . . . 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 8941	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) # 0 )
32	24, 16, 27, 31	divmulap3d 9119	. . . . . . 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 9081	. . . . . 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 8417	. . . 4 |- 0 < 1
39		0re 8290	. . . . 5 |- 0 e. RR
40		1re 8289	. . . . 5 |- 1 e. RR
41		axltwlin 8357	. . . . 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 11019	. . . 4 |- ( 0 ^ 2 ) = 0
46		oveq1 6065	. . . 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 11022	. . . 4 |- ( 1 ^ 2 ) = 1
50		oveq1 6065	. . . 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 4114  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   x. cmul 8148   < clt 8324   - cmin 8461   / cdiv 8966   2c2 9308   ^cexp 10927

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8463  df-neg 8464  df-reap 8867  df-ap 8874  df-div 8967  df-inn 9258  df-2 9316  df-n0 9517  df-z 9598  df-uz 9875  df-seqfrec 10837  df-exp 10928

This theorem is referenced by: (None)