Theorem resq01 11078

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 531	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A e. RR )
2	1	recnd 8348	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A e. CC )
3		sqval 11017	. . . . . . . . 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 2273	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A x. A ) = A )
7		simplr 533	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> 0 < A )
8	1, 7	gt0ap0d 8951	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A # 0 )
9	2, 2, 2, 8	divmulapd 9136	. . . . . . 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 9110	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> ( A / A ) = 1 )
12	10, 11	eqtr3d 2273	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( A ^ 2 ) = A ) -> A = 1 )
13	12	olcd 746	. . . 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 531	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A e. RR )
16	15	recnd 8348	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A e. CC )
17	16, 16	muls1d 8739	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. ( A - 1 ) ) = ( ( A x. A ) - A ) )
18	16, 16	mulcld 8340	. . . . . . . . 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 2273	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A x. A ) = A )
22	18, 21	subeq0bd 8700	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( ( A x. A ) - A ) = 0 )
23	17, 22	eqtr2d 2272	. . . . . . 7 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 0 = ( A x. ( A - 1 ) ) )
24		0cnd 8313	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 0 e. CC )
25		1red 8335	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> 1 e. RR )
26	15, 25	resubcld 8702	. . . . . . . . 9 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) e. RR )
27	26	recnd 8348	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) e. CC )
28		simplr 533	. . . . . . . . . 10 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A < 1 )
29	15, 25	sublt0d 8892	. . . . . . . . . 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 8971	. . . . . . . 8 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( A - 1 ) # 0 )
32	24, 16, 27, 31	divmulap3d 9149	. . . . . . 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 9111	. . . . . 6 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> ( 0 / ( A - 1 ) ) = 0 )
35	33, 34	eqtr3d 2273	. . . . 5 |- ( ( ( A e. RR /\ A < 1 ) /\ ( A ^ 2 ) = A ) -> A = 0 )
36	35	orcd 745	. . . 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 8447	. . . 4 |- 0 < 1
39		0re 8320	. . . . 5 |- 0 e. RR
40		1re 8319	. . . . 5 |- 1 e. RR
41		axltwlin 8387	. . . . 5 |- ( ( 0 e. RR /\ 1 e. RR /\ A e. RR ) -> ( 0 < 1 -> ( 0 < A \/ A < 1 ) ) )
42	39, 40, 41	mp3an12 1368	. . . 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 810	. 2 |- ( A e. RR -> ( ( A ^ 2 ) = A -> ( A = 0 \/ A = 1 ) ) )
45		sq0 11050	. . . 4 |- ( 0 ^ 2 ) = 0
46		oveq1 6086	. . . 4 |- ( A = 0 -> ( A ^ 2 ) = ( 0 ^ 2 ) )
47		id 19	. . . 4 |- ( A = 0 -> A = 0 )
48	45, 46, 47	3eqtr4a 2297	. . 3 |- ( A = 0 -> ( A ^ 2 ) = A )
49		sq1 11053	. . . 4 |- ( 1 ^ 2 ) = 1
50		oveq1 6086	. . . 4 |- ( A = 1 -> ( A ^ 2 ) = ( 1 ^ 2 ) )
51		id 19	. . . 4 |- ( A = 1 -> A = 1 )
52	49, 50, 51	3eqtr4a 2297	. . 3 |- ( A = 1 -> ( A ^ 2 ) = A )
53	48, 52	jaoi 728	. 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 720   = wceq 1402   e. wcel 2209   class class class wbr 4128  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   x. cmul 8178   < clt 8354   - cmin 8491   / cdiv 8996   2c2 9338   ^cexp 10958

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-n0 9547  df-z 9628  df-uz 9905  df-seqfrec 10868  df-exp 10959

This theorem is referenced by:  sq01  11643