Theorem ppidif 16175

Description: The difference of the prime-counting function π at two points counts the number of primes in an interval. (Contributed by Mario Carneiro, 21-Sep-2014.)

Assertion

Ref	Expression
ppidif	|- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )

Proof of Theorem ppidif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluzelz 9940	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
2		eluzel2 9935	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
3		2z 9676	. . . . . . 7 |- 2 e. ZZ
4		zmincl 12020	. . . . . . 7 |- ( ( M e. ZZ /\ 2 e. ZZ ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
5	2, 3, 4	sylancl 417	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) e. ZZ )
6	3	a1i 9	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> 2 e. ZZ )
7	2	zred 9772	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> M e. RR )
8		2re 9376	. . . . . . 7 |- 2 e. RR
9		min2inf 12014	. . . . . . 7 |- ( ( M e. RR /\ 2 e. RR ) -> inf ( { M , 2 } , RR , < ) <_ 2 )
10	7, 8, 9	sylancl 417	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) <_ 2 )
11		eluz2 9936	. . . . . 6 |- ( 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) <-> (inf ( { M , 2 } , RR , < ) e. ZZ /\ 2 e. ZZ /\ inf ( { M , 2 } , RR , < ) <_ 2 ) )
12	5, 6, 10, 11	syl3anbrc 1212	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) )
13		ppival2g 16162	. . . . 5 |- ( ( N e. ZZ /\ 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) ) -> (π ` N ) = (♯ ` ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) )
14	1, 12, 13	syl2anc 415	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (π ` N ) = (♯ ` ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) )
15		min1inf 12013	. . . . . . . . . . 11 |- ( ( M e. RR /\ 2 e. RR ) -> inf ( { M , 2 } , RR , < ) <_ M )
16	7, 8, 15	sylancl 417	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> inf ( { M , 2 } , RR , < ) <_ M )
17		eluz2 9936	. . . . . . . . . 10 |- ( M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) <-> (inf ( { M , 2 } , RR , < ) e. ZZ /\ M e. ZZ /\ inf ( { M , 2 } , RR , < ) <_ M ) )
18	5, 2, 16, 17	syl3anbrc 1212	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) )
19		id 19	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ( ZZ>= ` M ) )
20		elfzuzb 10432	. . . . . . . . 9 |- ( M e. (inf ( { M , 2 } , RR , < ) ... N ) <-> ( M e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) /\ N e. ( ZZ>= ` M ) ) )
21	18, 19, 20	sylanbrc 421	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M e. (inf ( { M , 2 } , RR , < ) ... N ) )
22		fzsplit 10466	. . . . . . . 8 |- ( M e. (inf ( { M , 2 } , RR , < ) ... N ) -> (inf ( { M , 2 } , RR , < ) ... N ) = ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) )
23	21, 22	syl 14	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... N ) = ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) )
24	23	ineq1d 3431	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) )
25		indir 3480	. . . . . 6 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) u. ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) )
26	24, 25	eqtrdi 2287	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) )
27	26	fveq2d 5699	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( (inf ( { M , 2 } , RR , < ) ... N ) i^i Prime ) ) = (♯ ` ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
28	5, 2	fzfigd 10881	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> (inf ( { M , 2 } , RR , < ) ... M ) e. Fin )
29		inss1 3451	. . . . . . 7 |- ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M )
30	29	a1i 9	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M ) )
31		animorrl 838	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> ( x e. (inf ( { M , 2 } , RR , < ) ... M ) \/ -. x e. (inf ( { M , 2 } , RR , < ) ... M ) ) )
32		df-dc 847	. . . . . . . . . 10 |- (DECID x e. (inf ( { M , 2 } , RR , < ) ... M ) <-> ( x e. (inf ( { M , 2 } , RR , < ) ... M ) \/ -. x e. (inf ( { M , 2 } , RR , < ) ... M ) ) )
33	31, 32	sylibr 134	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. (inf ( { M , 2 } , RR , < ) ... M ) )
34		elfzelz 10438	. . . . . . . . . . 11 |- ( x e. (inf ( { M , 2 } , RR , < ) ... M ) -> x e. ZZ )
35	34	adantl 277	. . . . . . . . . 10 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> x e. ZZ )
36		prmdcz 12925	. . . . . . . . . 10 |- ( x e. ZZ -> DECID x e. Prime )
37	35, 36	syl 14	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. Prime )
38	33, 37	dcand 945	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
39		elin 3412	. . . . . . . . 9 |- ( x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) <-> ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
40	39	dcbii 852	. . . . . . . 8 |- (DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) <-> DECID ( x e. (inf ( { M , 2 } , RR , < ) ... M ) /\ x e. Prime ) )
41	38, 40	sylibr 134	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ x e. (inf ( { M , 2 } , RR , < ) ... M ) ) -> DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
42	41	ralrimiva 2623	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> A. x e. (inf ( { M , 2 } , RR , < ) ... M )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) )
43		ssfidc 7245	. . . . . 6 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) e. Fin /\ ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) C_ (inf ( { M , 2 } , RR , < ) ... M ) /\ A. x e. (inf ( { M , 2 } , RR , < ) ... M )DECID x e. ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin )
44	28, 30, 42, 43	syl3anc 1278	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin )
45	2	peano2zd 9775	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( M + 1 ) e. ZZ )
46	45, 1	fzfigd 10881	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( M + 1 ) ... N ) e. Fin )
47		inss1 3451	. . . . . . 7 |- ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N )
48	47	a1i 9	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N ) )
49		orc 724	. . . . . . . . . . 11 |- ( x e. ( ( M + 1 ) ... N ) -> ( x e. ( ( M + 1 ) ... N ) \/ -. x e. ( ( M + 1 ) ... N ) ) )
50		df-dc 847	. . . . . . . . . . 11 |- (DECID x e. ( ( M + 1 ) ... N ) <-> ( x e. ( ( M + 1 ) ... N ) \/ -. x e. ( ( M + 1 ) ... N ) ) )
51	49, 50	sylibr 134	. . . . . . . . . 10 |- ( x e. ( ( M + 1 ) ... N ) -> DECID x e. ( ( M + 1 ) ... N ) )
52	51	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. ( ( M + 1 ) ... N ) )
53		elfzelz 10438	. . . . . . . . . . 11 |- ( x e. ( ( M + 1 ) ... N ) -> x e. ZZ )
54	53, 36	syl 14	. . . . . . . . . 10 |- ( x e. ( ( M + 1 ) ... N ) -> DECID x e. Prime )
55	54	adantl 277	. . . . . . . . 9 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. Prime )
56	52, 55	dcand 945	. . . . . . . 8 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
57		elin 3412	. . . . . . . . 9 |- ( x e. ( ( ( M + 1 ) ... N ) i^i Prime ) <-> ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
58	57	dcbii 852	. . . . . . . 8 |- (DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) <-> DECID ( x e. ( ( M + 1 ) ... N ) /\ x e. Prime ) )
59	56, 58	sylibr 134	. . . . . . 7 |- ( ( N e. ( ZZ>= ` M ) /\ x e. ( ( M + 1 ) ... N ) ) -> DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) )
60	59	ralrimiva 2623	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> A. x e. ( ( M + 1 ) ... N )DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) )
61		ssfidc 7245	. . . . . 6 |- ( ( ( ( M + 1 ) ... N ) e. Fin /\ ( ( ( M + 1 ) ... N ) i^i Prime ) C_ ( ( M + 1 ) ... N ) /\ A. x e. ( ( M + 1 ) ... N )DECID x e. ( ( ( M + 1 ) ... N ) i^i Prime ) ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin )
62	46, 48, 60, 61	syl3anc 1278	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin )
63	7	ltp1d 9262	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M < ( M + 1 ) )
64		fzdisj 10467	. . . . . . . 8 |- ( M < ( M + 1 ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
65	63, 64	syl 14	. . . . . . 7 |- ( N e. ( ZZ>= ` M ) -> ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) = (/) )
66	65	ineq1d 3431	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( (/) i^i Prime ) )
67		inindir 3449	. . . . . 6 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i ( ( M + 1 ) ... N ) ) i^i Prime ) = ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) )
68		0in 3558	. . . . . 6 |- ( (/) i^i Prime ) = (/)
69	66, 67, 68	3eqtr3g 2294	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) )
70		hashun 11259	. . . . 5 |- ( ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin /\ ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin /\ ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) i^i ( ( ( M + 1 ) ... N ) i^i Prime ) ) = (/) ) -> (♯ ` ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) = ( (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) + (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
71	44, 62, 69, 70	syl3anc 1278	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) u. ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) = ( (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) + (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
72	14, 27, 71	3eqtrd 2275	. . 3 |- ( N e. ( ZZ>= ` M ) -> (π ` N ) = ( (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) + (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) )
73		ppival2g 16162	. . . 4 |- ( ( M e. ZZ /\ 2 e. ( ZZ>= ` inf ( { M , 2 } , RR , < ) ) ) -> (π ` M ) = (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) )
74	2, 12, 73	syl2anc 415	. . 3 |- ( N e. ( ZZ>= ` M ) -> (π ` M ) = (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) )
75	72, 74	oveq12d 6103	. 2 |- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = ( ( (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) + (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) - (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) ) )
76		hashcl 11234	. . . . 5 |- ( ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) e. Fin -> (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) e. NN0 )
77	44, 76	syl 14	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) e. NN0 )
78	77	nn0cnd 9626	. . 3 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) e. CC )
79		hashcl 11234	. . . . 5 |- ( ( ( ( M + 1 ) ... N ) i^i Prime ) e. Fin -> (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. NN0 )
80	62, 79	syl 14	. . . 4 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. NN0 )
81	80	nn0cnd 9626	. . 3 |- ( N e. ( ZZ>= ` M ) -> (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) e. CC )
82	78, 81	pncan2d 8640	. 2 |- ( N e. ( ZZ>= ` M ) -> ( ( (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) + (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) ) - (♯ ` ( (inf ( { M , 2 } , RR , < ) ... M ) i^i Prime ) ) ) = (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )
83	75, 82	eqtrd 2271	1 |- ( N e. ( ZZ>= ` M ) -> ( (π ` N ) - (π ` M ) ) = (♯ ` ( ( ( M + 1 ) ... N ) i^i Prime ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {cpr 3710   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Fincfn 7022  infcinf 7323   RRcr 8178   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8498   2c2 9357   NN0cn0 9567   ZZcz 9648   ZZ>=cuz 9930   ...cfz 10421  ♯chash 11228   Primecprime 12901  πcppi 16152

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-stab 843  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-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-2o 6688  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  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-icc 10307  df-fz 10422  df-fl 10715  df-mod 10773  df-seqfrec 10898  df-exp 10989  df-ihash 11229  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-dvds 12571  df-prm 12902  df-ppi 16154

This theorem is used by:  ppiqub  16194