Theorem qdiff 16764

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 16763 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 9549	. . . . 5 |- 1 e. ZZ
2		zq 9904	. . . . 5 |- ( 1 e. ZZ -> 1 e. QQ )
3	1, 2	ax-mp 5	. . . 4 |- 1 e. QQ
4		qsubcl 9916	. . . 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 9913	. . . . 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 9903	. . . . . . . . . 10 |- ( A e. QQ -> A e. RR )
9		1rp 9936	. . . . . . . . . . . . 13 |- 1 e. RR+
10	9, 9	pm3.2i 272	. . . . . . . . . . . 12 |- ( 1 e. RR+ /\ 1 e. RR+ )
11		rpaddcl 9956	. . . . . . . . . . . 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 10009	. . . . . . . . . 10 |- ( A e. QQ -> A < ( A + ( 1 + 1 ) ) )
14	8, 13	ltned 8335	. . . . . . . . 9 |- ( A e. QQ -> A =/= ( A + ( 1 + 1 ) ) )
15	14	neneqd 2424	. . . . . . . 8 |- ( A e. QQ -> -. A = ( A + ( 1 + 1 ) ) )
16	15	neqcomd 2236	. . . . . . 7 |- ( A e. QQ -> -. ( A + ( 1 + 1 ) ) = A )
17		qcn 9912	. . . . . . . . 9 |- ( A e. QQ -> A e. CC )
18		1cnd 8238	. . . . . . . . 9 |- ( A e. QQ -> 1 e. CC )
19	17, 18, 18	addassd 8244	. . . . . . . 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 680	. . . . . 6 |- ( A e. QQ -> -. ( ( A + 1 ) + 1 ) = A )
22	17, 18	addcld 8241	. . . . . . 7 |- ( A e. QQ -> ( A + 1 ) e. CC )
23	17, 18, 22	subadd2d 8551	. . . . . 6 |- ( A e. QQ -> ( ( A - 1 ) = ( A + 1 ) <-> ( ( A + 1 ) + 1 ) = A ) )
24	21, 23	mtbird 680	. . . . 5 |- ( A e. QQ -> -. ( A - 1 ) = ( A + 1 ) )
25	24	neqned 2410	. . . 4 |- ( A e. QQ -> ( A - 1 ) =/= ( A + 1 ) )
26	18	absnegd 11812	. . . . 5 |- ( A e. QQ -> ( abs ` -u 1 ) = ( abs ` 1 ) )
27	17, 17, 18	subsub4d 8563	. . . . . . 7 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( A - ( A + 1 ) ) )
28	17	subidd 8520	. . . . . . . . 9 |- ( A e. QQ -> ( A - A ) = 0 )
29	28	oveq1d 6043	. . . . . . . 8 |- ( A e. QQ -> ( ( A - A ) - 1 ) = ( 0 - 1 ) )
30		df-neg 8395	. . . . . . . 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 5652	. . . . 5 |- ( A e. QQ -> ( abs ` ( A - ( A + 1 ) ) ) = ( abs ` -u 1 ) )
34	17, 18	nncand 8537	. . . . . 6 |- ( A e. QQ -> ( A - ( A - 1 ) ) = 1 )
35	34	fveq2d 5652	. . . . 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 2417	. . . . . 6 |- ( r = ( A + 1 ) -> ( ( A - 1 ) =/= r <-> ( A - 1 ) =/= ( A + 1 ) ) )
38		oveq2 6036	. . . . . . . 8 |- ( r = ( A + 1 ) -> ( A - r ) = ( A - ( A + 1 ) ) )
39	38	fveq2d 5652	. . . . . . 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 2911	. . . 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 1272	. . 3 |- ( A e. QQ -> E. r e. QQ ( ( A - 1 ) =/= r /\ ( abs ` ( A - ( A - 1 ) ) ) = ( abs ` ( A - r ) ) ) )
44		neeq1 2416	. . . . . 6 |- ( q = ( A - 1 ) -> ( q =/= r <-> ( A - 1 ) =/= r ) )
45		oveq2 6036	. . . . . . 7 |- ( q = ( A - 1 ) -> ( A - q ) = ( A - ( A - 1 ) ) )
46	45	fveqeq2d 5656	. . . . . 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 2534	. . . 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 2911	. . 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 9258	. . . . . . . . . 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 8250	. . . . . . . . . . 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 9903	. . . . . . . . . . . . 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 8250	. . . . . . . . . . 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 8242	. . . . . . . . . 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 9903	. . . . . . . . . . . . 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 8250	. . . . . . . . . . 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 8242	. . . . . . . . . 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 8635	. . . . . . . . 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 10979	. . . . . . . . . 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 8242	. . . . . . . . . 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 8242	. . . . . . . . . 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 8566	. . . . . . . . 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 8602	. . . . . . . . . . . . 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 8602	. . . . . . . . . . . . 13 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( A - r ) e. RR )
72		sqabs 11705	. . . . . . . . . . . . 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 10961	. . . . . . . . . . . 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 10961	. . . . . . . . . . . 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 8532	. . . . . . . . . . 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 10979	. . . . . . . . . . 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 8532	. . . . . . . . . . 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 10979	. . . . . . . . . . 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 8583	. . . . . . . . . 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 8532	. . . . . . . . 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 8532	. . . . . . . . 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 9278	. . . . . . . . . 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 9034	. . . . . . . 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 8635	. . . . . . 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 9431	. . . . . . 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 8532	. . . . . . 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 8541	. . . . . . . 8 |- ( ( ( A e. RR /\ ( q e. QQ /\ r e. QQ ) ) /\ ( q =/= r /\ ( abs ` ( A - q ) ) = ( abs ` ( A - r ) ) ) ) -> ( q - r ) =/= 0 )
99		qsubcl 9916	. . . . . . . . . 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 9534	. . . . . . . . . 10 |- 0 e. ZZ
102		zq 9904	. . . . . . . . . 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 9917	. . . . . . . . 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 9047	. . . . . 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 10919	. . . . . . . . 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 10919	. . . . . . . . 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 9916	. . . . . . . 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 9551	. . . . . . . 8 |- 2 e. ZZ
116		zq 9904	. . . . . . . 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 9277	. . . . . . . 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 9921	. . . . . . 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 1274	. . . . . 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 9921	. . . . . 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 1274	. . . . 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 2659	. 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 1398   e. wcel 2202   =/= wne 2403   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8073   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   - cmin 8392   -ucneg 8393   # cap 8803   / cdiv 8894   2c2 9236   ZZcz 9523   QQcq 9897   RR+crp 9932   ^cexp 10846   abscabs 11620

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622

This theorem is referenced by: (None)