Theorem qdiff 16679

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 16678 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 9505	. . . . 5 |- 1 e. ZZ
2		zq 9860	. . . . 5 |- ( 1 e. ZZ -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		qsubcl 9872	. . . 4 |- ( ( A e. QQ /\ 1 e. QQ ) -> ( A - 1 ) e. QQ )
5	3, 4	mpan2 425	. . 3 |- ( A e. QQ -> ( A - 1 ) e. QQ )
6		qaddcl 9869	. . . . 5 |- ( ( A e. QQ /\ 1 e. QQ ) -> ( A + 1 ) e. QQ )
7	3, 6	mpan2 425	. . . 4 |- ( A e. QQ -> ( A + 1 ) e. QQ )
8		qre 9859	. . . . . . . . . 10 |- ( A e. QQ -> A e. RR )
9		1rp 9892	. . . . . . . . . . . . 13 |- 1 e. RR+
10	9, 9	pm3.2i 272	. . . . . . . . . . . 12 |- ( 1 e. RR+ /\ 1 e. RR+ )
11		rpaddcl 9912	. . . . . . . . . . . 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 9965	. . . . . . . . . 10 |- ( A e. QQ -> A < ( A + ( 1 + 1 ) ) )
14	8, 13	ltned 8293	. . . . . . . . 9 |- ( A e. QQ -> A =/= ( A + ( 1 + 1 ) ) )
15	14	neneqd 2423	. . . . . . . 8 |- ( A e. QQ -> -. A = ( A + ( 1 + 1 ) ) )
16	15	neqcomd 2236	. . . . . . 7 |- ( A e. QQ -> -. ( A + ( 1 + 1 ) ) = A )
17		qcn 9868	. . . . . . . . 9 |- ( A e. QQ -> A e. CC )
18		1cnd 8195	. . . . . . . . 9 |- ( A e. QQ -> 1 e. CC )
19	17, 18, 18	addassd 8202	. . . . . . . 8 |- ( A e. QQ -> ( ( A + 1 ) + 1 ) = ( A + ( 1 + 1 ) ) )
20	19	eqeq1d 2240	. . . . . . 7 |- ( A e. QQ -> ( ( ( A + 1 ) + 1 ) = A <-> ( A + ( 1 + 1 ) ) = A ) )
21	16, 20	mtbird 679	. . . . . 6 |- ( A e. QQ -> -. ( ( A + 1 ) + 1 ) = A )
22	17, 18	addcld 8199	. . . . . . 7 |- ( A e. QQ -> ( A + 1 ) e. CC )
23	17, 18, 22	subadd2d 8509	. . . . . 6 |- ( A e. QQ -> ( ( A - 1 ) = ( A + 1 ) <-> ( ( A + 1 ) + 1 ) = A ) )
24	21, 23	mtbird 679	. . . . 5 |- ( A e. QQ -> -. ( A - 1 ) = ( A + 1 ) )
25	24	neqned 2409	. . . 4 |- ( A e. QQ -> ( A - 1 ) =/= ( A + 1 ) )
26	18	absnegd 11754	. . . . 5 |- ( A e. QQ -> ( abs ` -u 1 ) = ( abs ` 1 ) )
27	17, 17, 18	subsub4d 8521	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( A - ( A + 1 ) ) )
28	17	subidd 8478	. . . . . . . . 9 |- ( A e. QQ -> ( A - A ) = 0 )
29	28	oveq1d 6033	. . . . . . . 8 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( 0 - 1 ) )
30		df-neg 8353	. . . . . . . 8 |- -u 1 = ( 0 - 1 )
31	29, 30	eqtr4di 2282	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = -u 1 )
32	27, 31	eqtr3d 2266	. . . . . 6 |- ( A e. QQ -> ( A - ( A + 1 ) ) = -u 1 )
33	32	fveq2d 5643	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A + 1 ) ) ) = ( abs ` -u 1 ) )
34	17, 18	nncand 8495	. . . . . 6 |- ( A e. QQ -> ( A - ( A - 1 ) ) = 1 )
35	34	fveq2d 5643	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` 1 ) )
36	26, 33, 35	3eqtr4rd 2275	. . . 4 |- ( A e. QQ -> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) )
37		neeq2 2416	. . . . . 6 |- ( r = ( A + 1 ) -> ( ( A - 1 ) =/= r <-> ( A - 1 ) =/= ( A + 1 ) ) )
38		oveq2 6026	. . . . . . . 8 |- ( r = ( A + 1 ) -> ( A - r ) = ( A - ( A + 1 ) ) )
39	38	fveq2d 5643	. . . . . . 7 |- ( r = ( A + 1 ) -> ( abs ` ( A - r ) ) = ( abs ` ( A - ( A + 1 ) ) ) )
40	39	eqeq2d 2243	. . . . . 6 |- ( r = ( A + 1 ) -> ( ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) <-> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - ( A + 1 ) ) ) ) )
41	37, 40	anbi12d 473	. . . . 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 2910	. . . 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 1271	. . 3 |- ( A e. QQ -> E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
44		neeq1 2415	. . . . . 6 |- ( q = ( A - 1 ) -> ( q =/= r <-> ( A - 1 ) =/= r ) )
45		oveq2 6026	. . . . . . 7 |- ( q = ( A - 1 ) -> ( A - q ) = ( A - ( A - 1 ) ) )
46	45	fveqeq2d 5647	. . . . . 6 |- ( q = ( A - 1 ) -> ( ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) <-> ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
47	44, 46	anbi12d 473	. . . . 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 2533	. . . 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 2910	. . 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 411	. 2 |- ( A e. QQ -> E. q e. QQ E. r e. QQ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) )
51		2cnd 9216	. . . . . . . . . 10 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> 2 e. CC )
52		simpll 527	. . . . . . . . . . . 12 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> A e. RR )
53	52	recnd 8208	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> A e. CC )
54		simplrl 537	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> q e. QQ )
55		qre 9859	. . . . . . . . . . . . 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 8208	. . . . . . . . . . 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 8200	. . . . . . . . . 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 538	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> r e. QQ )
60		qre 9859	. . . . . . . . . . . . 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 8208	. . . . . . . . . . 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 8200	. . . . . . . . . 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 8593	. . . . . . . . 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 10934	. . . . . . . . . 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 8200	. . . . . . . . . 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 8200	. . . . . . . . . 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 8524	. . . . . . . . 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 533	. . . . . . . . . . . 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 8560	. . . . . . . . . . . . 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 8560	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A - r ) e. RR )
72		sqabs 11647	. . . . . . . . . . . . 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 411	. . . . . . . . . . . 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 10916	. . . . . . . . . . . 12 |- ( ( A e. CC /\ q e. CC ) -> ( ( A - q ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. q ) ) ) + ( q ^ 2 ) ) )
76	53, 57, 75	syl2anc 411	. . . . . . . . . . 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 10916	. . . . . . . . . . . 12 |- ( ( A e. CC /\ r e. CC ) -> ( ( A - r ) ^ 2 ) = ( ( ( A ^ 2 ) - ( 2 x. ( A x. r ) ) ) + ( r ^ 2 ) ) )
78	53, 62, 77	syl2anc 411	. . . . . . . . . . 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 2272	. . . . . . . . . 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 8490	. . . . . . . . . . 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 10934	. . . . . . . . . . 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 8490	. . . . . . . . . . 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 10934	. . . . . . . . . . 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 8541	. . . . . . . . . 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 2270	. . . . . . . 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 8490	. . . . . . . . 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 8490	. . . . . . . . 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 9236	. . . . . . . . . 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 8992	. . . . . . . 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 8593	. . . . . . 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 2267	. . . . . 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 9389	. . . . . . 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 8490	. . . . . . 7 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) e. CC )
97		simprl 531	. . . . . . . . 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 8499	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) =/= 0 )
99		qsubcl 9872	. . . . . . . . . 10 |- ( ( q e. QQ /\ r e. QQ ) -> ( q - r ) e. QQ )
100	54, 59, 99	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) e. QQ )
101		0z 9490	. . . . . . . . . 10 |- 0 e. ZZ
102		zq 9860	. . . . . . . . . 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 9873	. . . . . . . . 9 |- ( ( ( q - r ) e. QQ /\ 0 e. QQ ) -> ( ( q - r ) # 0 <-> ( q - r ) =/= 0 ) )
105	100, 103, 104	syl2anc 411	. . . . . . . 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 9005	. . . . . 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 10874	. . . . . . . . 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 10874	. . . . . . . . 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 9872	. . . . . . . 8 |- ( ( ( q ^ 2 ) e. QQ /\ ( r ^ 2 ) e. QQ ) -> ( ( q ^ 2 ) - ( r ^ 2 ) ) e. QQ )
114	110, 112, 113	syl2anc 411	. . . . . . 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 9507	. . . . . . . 8 |- 2 e. ZZ
116		zq 9860	. . . . . . . 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 9235	. . . . . . . 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 9877	. . . . . . 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 1273	. . . . . 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 9877	. . . . . 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 1273	. . . . 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 2309	. . . 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 2658	. 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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   =/= wne 2402   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   - cmin 8350   -ucneg 8351   # cap 8761   / cdiv 8852   2c2 9194   ZZcz 9479   QQcq 9853   RR+crp 9888   ^cexp 10801   abscabs 11562

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564

This theorem is referenced by: (None)