Theorem qdiff 17305

Description: The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to apdiff 17304 but by stating the result positively we can completely sidestep the issue of not equal versus apart in the statement of the result. From an online post by Ingo Blechschmidt. (Contributed by Jim Kingdon, 24-Apr-2026.)

Assertion

Ref	Expression
qdiff	|- ( A e. RR -> ( A e. QQ <-> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) )

Distinct variable group:   A, q, r

Proof of Theorem qdiff

Step	Hyp	Ref	Expression
1		1z 9681	. . . . 5 |- 1 e. ZZ
2		zq 10042	. . . . 5 |- ( 1 e. ZZ -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		qsubcl 10054	. . . 4 |- ( ( A e. QQ /\ 1 e. QQ ) -> ( A - 1 ) e. QQ )
5	3, 4	mpan2 429	. . 3 |- ( A e. QQ -> ( A - 1 ) e. QQ )
6		qaddcl 10051	. . . . 5 |- ( ( A e. QQ /\ 1 e. QQ ) -> ( A + 1 ) e. QQ )
7	3, 6	mpan2 429	. . . 4 |- ( A e. QQ -> ( A + 1 ) e. QQ )
8		qre 10041	. . . . . . . . . 10 |- ( A e. QQ -> A e. RR )
9		1rp 10075	. . . . . . . . . . . . 13 |- 1 e. RR+
10	9, 9	pm3.2i 272	. . . . . . . . . . . 12 |- ( 1 e. RR+ /\ 1 e. RR+ )
11		rpaddcl 10095	. . . . . . . . . . . 12 |- ( ( 1 e. RR+ /\ 1 e. RR+ ) -> ( 1 + 1 ) e. RR+ )
12	10, 11	mp1i 10	. . . . . . . . . . 11 |- ( A e. QQ -> ( 1 + 1 ) e. RR+ )
13	8, 12	ltaddrpd 10148	. . . . . . . . . 10 |- ( A e. QQ -> A < ( A + ( 1 + 1 ) ) )
14	8, 13	ltned 8446	. . . . . . . . 9 |- ( A e. QQ -> A =/= ( A + ( 1 + 1 ) ) )
15	14	neneqd 2441	. . . . . . . 8 |- ( A e. QQ -> -. A = ( A + ( 1 + 1 ) ) )
16	15	neqcomd 2243	. . . . . . 7 |- ( A e. QQ -> -. ( A + ( 1 + 1 ) ) = A )
17		qcn 10050	. . . . . . . . 9 |- ( A e. QQ -> A e. CC )
18		1cnd 8348	. . . . . . . . 9 |- ( A e. QQ -> 1 e. CC )
19	17, 18, 18	addassd 8354	. . . . . . . 8 |- ( A e. QQ -> ( ( A + 1 ) + 1 ) = ( A + ( 1 + 1 ) ) )
20	19	eqeq1d 2247	. . . . . . 7 |- ( A e. QQ -> ( ( ( A + 1 ) + 1 ) = A <-> ( A + ( 1 + 1 ) ) = A ) )
21	16, 20	mtbird 684	. . . . . 6 |- ( A e. QQ -> -. ( ( A + 1 ) + 1 ) = A )
22	17, 18	addcld 8351	. . . . . . 7 |- ( A e. QQ -> ( A + 1 ) e. CC )
23	17, 18, 22	subadd2d 8663	. . . . . 6 |- ( A e. QQ -> ( ( A - 1 ) = ( A + 1 ) <-> ( ( A + 1 ) + 1 ) = A ) )
24	21, 23	mtbird 684	. . . . 5 |- ( A e. QQ -> -. ( A - 1 ) = ( A + 1 ) )
25	24	neqned 2427	. . . 4 |- ( A e. QQ -> ( A - 1 ) =/= ( A + 1 ) )
26	18	absnegd 11978	. . . . 5 |- ( A e. QQ -> ( abs ` -u 1 ) = ( abs ` 1 ) )
27	17, 17, 18	subsub4d 8675	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( A - ( A + 1 ) ) )
28	17	subidd 8632	. . . . . . . . 9 |- ( A e. QQ -> ( A - A ) = 0 )
29	28	oveq1d 6103	. . . . . . . 8 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( 0 - 1 ) )
30		df-neg 8507	. . . . . . . 8 |- -u 1 = ( 0 - 1 )
31	29, 30	eqtr4di 2289	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = -u 1 )
32	27, 31	eqtr3d 2273	. . . . . 6 |- ( A e. QQ -> ( A - ( A + 1 ) ) = -u 1 )
33	32	fveq2d 5702	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A + 1 ) ) ) = ( abs ` -u 1 ) )
34	17, 18	nncand 8649	. . . . . 6 |- ( A e. QQ -> ( A - ( A - 1 ) ) = 1 )
35	34	fveq2d 5702	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` 1 ) )
36	26, 33, 35	3eqtr4rd 2282	. . . 4 |- ( A e. QQ -> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) )
37		neeq2 2434	. . . . . 6 |- ( r = ( A + 1 ) -> ( ( A - 1 ) =/= r <-> ( A - 1 ) =/= ( A + 1 ) ) )
38		oveq2 6096	. . . . . . . 8 |- ( r = ( A + 1 ) -> ( A - r ) = ( A - ( A + 1 ) ) )
39	38	fveq2d 5702	. . . . . . 7 |- ( r = ( A + 1 ) -> ( abs ` ( A - r ) ) = ( abs ` ( A - ( A + 1 ) ) ) )
40	39	eqeq2d 2250	. . . . . 6 |- ( r = ( A + 1 ) -> ( ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) <-> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) ) )
41	37, 40	anbi12d 477	. . . . 5 |- ( r = ( A + 1 ) -> ( ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) <-> ( ( A - 1 ) =/= ( A + 1 ) /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) ) ) )
42	41	rspcev 2929	. . . 4 |- ( ( ( A + 1 ) e. QQ /\ ( ( A - 1 ) =/= ( A + 1 ) /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) ) ) -> E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
43	7, 25, 36, 42	syl12anc 1276	. . 3 |- ( A e. QQ -> E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
44		neeq1 2433	. . . . . 6 |- ( q = ( A - 1 ) -> ( q =/= r <-> ( A - 1 ) =/= r ) )
45		oveq2 6096	. . . . . . 7 |- ( q = ( A - 1 ) -> ( A - q ) = ( A - ( A - 1 ) ) )
46	45	fveqeq2d 5706	. . . . . 6 |- ( q = ( A - 1 ) -> ( ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) <-> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
47	44, 46	anbi12d 477	. . . . 5 |- ( q = ( A - 1 ) -> ( ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) <-> ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) ) )
48	47	rexbidv 2551	. . . 4 |- ( q = ( A - 1 ) -> ( E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) <-> E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) ) )
49	48	rspcev 2929	. . 3 |- ( ( ( A - 1 ) e. QQ /\ E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) ) -> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) )
50	5, 43, 49	syl2anc 415	. 2 |- ( A e. QQ -> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) )
51		2cnd 9386	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 2 e. CC )
52		simpll 531	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> A e. RR )
53	52	recnd 8360	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> A e. CC )
54		simplrl 541	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> q e. QQ )
55		qre 10041	. . . . . . . . . . . . 13 |- ( q e. QQ -> q e. RR )
56	54, 55	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> q e. RR )
57	56	recnd 8360	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> q e. CC )
58	53, 57	mulcld 8352	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A x. q ) e. CC )
59		simplrr 542	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> r e. QQ )
60		qre 10041	. . . . . . . . . . . . 13 |- ( r e. QQ -> r e. RR )
61	59, 60	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> r e. RR )
62	61	recnd 8360	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> r e. CC )
63	53, 62	mulcld 8352	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A x. r ) e. CC )
64	51, 58, 63	subdid 8748	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( 2 x. ( ( A x. q ) - ( A x. r ) ) ) = ( ( 2 x. ( A x. q ) ) - ( 2 x. ( A x. r ) ) ) )
65	53	sqcld 11130	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A ^ 2 ) e. CC )
66	51, 63	mulcld 8352	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( 2 x. ( A x. r ) ) e. CC )
67	51, 58	mulcld 8352	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( 2 x. ( A x. q ) ) e. CC )
68	65, 66, 67	nnncan1d 8678	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) - ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) ) = ( ( 2 x. ( A x. q ) ) - ( 2 x. ( A x. r ) ) ) )
69		simprr 537	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) )
70	52, 56	resubcld 8715	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A - q ) e. RR )
71	52, 61	resubcld 8715	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A - r ) e. RR )
72		sqabs 11871	. . . . . . . . . . . . 13 |- ( ( ( A - q ) e. RR /\ ( A - r ) e. RR ) -> ( ( ( A - q ) ^ 2 ) = ( ( A - r ) ^ 2 ) <-> ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) )
73	70, 71, 72	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( A - q ) ^ 2 ) = ( ( A - r ) ^ 2 ) <-> ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) )
74	69, 73	mpbird 167	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A - q ) ^ 2 ) = ( ( A - r ) ^ 2 ) )
75		binom2sub 11111	. . . . . . . . . . . 12 |- ( ( A e. CC /\ q e. CC ) -> ( ( A - q ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) + ( q ^ 2 ) ) )
76	53, 57, 75	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A - q ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) + ( q ^ 2 ) ) )
77		binom2sub 11111	. . . . . . . . . . . 12 |- ( ( A e. CC /\ r e. CC ) -> ( ( A - r ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) + ( r ^ 2 ) ) )
78	53, 62, 77	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A - r ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) + ( r ^ 2 ) ) )
79	74, 76, 78	3eqtr3d 2279	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) + ( q ^ 2 ) ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) + ( r ^ 2 ) ) )
80	65, 67	subcld 8644	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) e. CC )
81	57	sqcld 11130	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q ^ 2 ) e. CC )
82	65, 66	subcld 8644	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) e. CC )
83	62	sqcld 11130	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( r ^ 2 ) e. CC )
84	80, 81, 82, 83	addsubeq4d 8695	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) + ( q ^ 2 ) ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) + ( r ^ 2 ) ) <-> ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) - ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) ) = ( ( q ^ 2 ) - ( r ^ 2 ) ) ) )
85	79, 84	mpbid 147	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) - ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) ) = ( ( q ^ 2 ) - ( r ^ 2 ) ) )
86	64, 68, 85	3eqtr2d 2277	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( 2 x. ( ( A x. q ) - ( A x. r ) ) ) = ( ( q ^ 2 ) - ( r ^ 2 ) ) )
87	81, 83	subcld 8644	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( q ^ 2 ) - ( r ^ 2 ) ) e. CC )
88	58, 63	subcld 8644	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( A x. q ) - ( A x. r ) ) e. CC )
89		2ap0 9406	. . . . . . . . . 10 |- 2 # 0
90	89	a1i 9	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 2 # 0 )
91	87, 51, 88, 90	divmulapd 9150	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) = ( ( A x. q ) - ( A x. r ) ) <-> ( 2 x. ( ( A x. q ) - ( A x. r ) ) ) = ( ( q ^ 2 ) - ( r ^ 2 ) ) ) )
92	86, 91	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) = ( ( A x. q ) - ( A x. r ) ) )
93	53, 57, 62	subdid 8748	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A x. ( q - r ) ) = ( ( A x. q ) - ( A x. r ) ) )
94	92, 93	eqtr4d 2274	. . . . . 6 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) = ( A x. ( q - r ) ) )
95	87	halfcld 9561	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) e. CC )
96	57, 62	subcld 8644	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) e. CC )
97		simprl 535	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> q =/= r )
98	57, 62, 97	subne0d 8653	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) =/= 0 )
99		qsubcl 10054	. . . . . . . . . 10 |- ( ( q e. QQ /\ r e. QQ ) -> ( q - r ) e. QQ )
100	54, 59, 99	syl2anc 415	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) e. QQ )
101		0z 9666	. . . . . . . . . 10 |- 0 e. ZZ
102		zq 10042	. . . . . . . . . 10 |- ( 0 e. ZZ -> 0 e. QQ )
103	101, 102	mp1i 10	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 0 e. QQ )
104		qapne 10055	. . . . . . . . 9 |- ( ( ( q - r ) e. QQ /\ 0 e. QQ ) -> ( ( q - r ) # 0 <-> ( q - r ) =/= 0 ) )
105	100, 103, 104	syl2anc 415	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( q - r ) # 0 <-> ( q - r ) =/= 0 ) )
106	98, 105	mpbird 167	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) # 0 )
107	95, 53, 96, 106	divmulap3d 9163	. . . . . 6 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) / ( q - r ) ) = A <-> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) = ( A x. ( q - r ) ) ) )
108	94, 107	mpbird 167	. . . . 5 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) / ( q - r ) ) = A )
109		qsqcl 11069	. . . . . . . . 9 |- ( q e. QQ -> ( q ^ 2 ) e. QQ )
110	54, 109	syl 14	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q ^ 2 ) e. QQ )
111		qsqcl 11069	. . . . . . . . 9 |- ( r e. QQ -> ( r ^ 2 ) e. QQ )
112	59, 111	syl 14	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( r ^ 2 ) e. QQ )
113		qsubcl 10054	. . . . . . . 8 |- ( ( ( q ^ 2 ) e. QQ /\ ( r ^ 2 ) e. QQ ) -> ( ( q ^ 2 ) - ( r ^ 2 ) ) e. QQ )
114	110, 112, 113	syl2anc 415	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( q ^ 2 ) - ( r ^ 2 ) ) e. QQ )
115		2z 9683	. . . . . . . 8 |- 2 e. ZZ
116		zq 10042	. . . . . . . 8 |- ( 2 e. ZZ -> 2 e. QQ )
117	115, 116	mp1i 10	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 2 e. QQ )
118		2ne0 9405	. . . . . . . 8 |- 2 =/= 0
119	118	a1i 9	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 2 =/= 0 )
120		qdivcl 10059	. . . . . . 7 |- ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) e. QQ /\ 2 e. QQ /\ 2 =/= 0 ) -> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) e. QQ )
121	114, 117, 119, 120	syl3anc 1278	. . . . . 6 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) e. QQ )
122		qdivcl 10059	. . . . . 6 |- ( ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) e. QQ /\ ( q - r ) e. QQ /\ ( q - r ) =/= 0 ) -> ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) / ( q - r ) ) e. QQ )
123	121, 100, 98, 122	syl3anc 1278	. . . . 5 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( ( ( ( q ^ 2 ) - ( r ^ 2 ) ) / 2 ) / ( q - r ) ) e. QQ )
124	108, 123	eqeltrrd 2316	. . . 4 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> A e. QQ )
125	124	ex 115	. . 3 |- ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) -> ( ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) -> A e. QQ ) )
126	125	rexlimdvva 2676	. 2 |- ( A e. RR -> ( E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) -> A e. QQ ) )
127	50, 126	impbid2 143	1 |- ( A e. RR -> ( A e. QQ <-> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   - cmin 8504   -ucneg 8505   # cap 8917   / cdiv 9010   2c2 9364   ZZcz 9655   QQcq 10035   RR+crp 10071   ^cexp 10996   abscabs 11785

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787

This theorem is used by: (None)