Theorem qdiff 17196

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 17195 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 9674	. . . . 5 |- 1 e. ZZ
2		zq 10035	. . . . 5 |- ( 1 e. ZZ -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		qsubcl 10047	. . . 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 10044	. . . . 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 10034	. . . . . . . . . 10 |- ( A e. QQ -> A e. RR )
9		1rp 10068	. . . . . . . . . . . . 13 |- 1 e. RR+
10	9, 9	pm3.2i 272	. . . . . . . . . . . 12 |- ( 1 e. RR+ /\ 1 e. RR+ )
11		rpaddcl 10088	. . . . . . . . . . . 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 10141	. . . . . . . . . 10 |- ( A e. QQ -> A < ( A + ( 1 + 1 ) ) )
14	8, 13	ltned 8440	. . . . . . . . 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 10043	. . . . . . . . 9 |- ( A e. QQ -> A e. CC )
18		1cnd 8342	. . . . . . . . 9 |- ( A e. QQ -> 1 e. CC )
19	17, 18, 18	addassd 8348	. . . . . . . 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 8345	. . . . . . 7 |- ( A e. QQ -> ( A + 1 ) e. CC )
23	17, 18, 22	subadd2d 8657	. . . . . 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 11970	. . . . 5 |- ( A e. QQ -> ( abs ` -u 1 ) = ( abs ` 1 ) )
27	17, 17, 18	subsub4d 8669	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( A - ( A + 1 ) ) )
28	17	subidd 8626	. . . . . . . . 9 |- ( A e. QQ -> ( A - A ) = 0 )
29	28	oveq1d 6100	. . . . . . . 8 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( 0 - 1 ) )
30		df-neg 8501	. . . . . . . 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 5699	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A + 1 ) ) ) = ( abs ` -u 1 ) )
34	17, 18	nncand 8643	. . . . . 6 |- ( A e. QQ -> ( A - ( A - 1 ) ) = 1 )
35	34	fveq2d 5699	. . . . 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 6093	. . . . . . . 8 |- ( r = ( A + 1 ) -> ( A - r ) = ( A - ( A + 1 ) ) )
39	38	fveq2d 5699	. . . . . . 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 6093	. . . . . . 7 |- ( q = ( A - 1 ) -> ( A - q ) = ( A - ( A - 1 ) ) )
46	45	fveqeq2d 5703	. . . . . 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 9379	. . . . . . . . . 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 8354	. . . . . . . . . . 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 10034	. . . . . . . . . . . . 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 8354	. . . . . . . . . . 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 8346	. . . . . . . . . 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 10034	. . . . . . . . . . . . 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 8354	. . . . . . . . . . 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 8346	. . . . . . . . . 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 8742	. . . . . . . . 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 11122	. . . . . . . . . 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 8346	. . . . . . . . . 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 8346	. . . . . . . . . 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 8672	. . . . . . . . 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 8709	. . . . . . . . . . . . 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 8709	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A - r ) e. RR )
72		sqabs 11863	. . . . . . . . . . . . 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 11103	. . . . . . . . . . . 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 11103	. . . . . . . . . . . 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 8638	. . . . . . . . . . 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 11122	. . . . . . . . . . 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 8638	. . . . . . . . . . 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 11122	. . . . . . . . . . 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 8689	. . . . . . . . . 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 8638	. . . . . . . . 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 8638	. . . . . . . . 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 9399	. . . . . . . . . 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 9144	. . . . . . . 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 8742	. . . . . . 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 9554	. . . . . . 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 8638	. . . . . . 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 8647	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) =/= 0 )
99		qsubcl 10047	. . . . . . . . . 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 9659	. . . . . . . . . 10 |- 0 e. ZZ
102		zq 10035	. . . . . . . . . 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 10048	. . . . . . . . 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 9157	. . . . . 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 11061	. . . . . . . . 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 11061	. . . . . . . . 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 10047	. . . . . . . 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 9676	. . . . . . . 8 |- 2 e. ZZ
116		zq 10035	. . . . . . . 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 9398	. . . . . . . 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 10052	. . . . . . 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 10052	. . . . . 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 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8498   -ucneg 8499   # cap 8911   / cdiv 9004   2c2 9357   ZZcz 9648   QQcq 10028   RR+crp 10064   ^cexp 10988   abscabs 11777

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779

This theorem is used by: (None)