Theorem addlocpr 7344

Description: Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 7311 to both A and B, and uses nqtri3or 7204 rather than prloc 7299 to decide whether q is too big to be in the lower cut of A +P. B (and deduce that if it is, then r must be in the upper cut). What the two proofs have in common is that they take the difference between q and r to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)

Assertion

Ref	Expression
addlocpr	|- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )

Distinct variable groups:   A, q, r   B, q, r

Proof of Theorem addlocpr

Dummy variables d e h p t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqq 7216	. . . . . 6 |- ( ( q e. Q. /\ r e. Q. ) -> ( q <Q r <-> E. p e. Q. ( q +Q p ) = r ) )
2	1	biimpa 294	. . . . 5 |- ( ( ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> E. p e. Q. ( q +Q p ) = r )
3	2	3adant1 999	. . . 4 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> E. p e. Q. ( q +Q p ) = r )
4		halfnqq 7218	. . . . . 6 |- ( p e. Q. -> E. h e. Q. ( h +Q h ) = p )
5	4	ad2antrl 481	. . . . 5 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) -> E. h e. Q. ( h +Q h ) = p )
6		prop 7283	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7311	. . . . . . . . . 10 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
8	6, 7	sylan 281	. . . . . . . . 9 |- ( ( A e. P. /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
9	8	adantlr 468	. . . . . . . 8 |- ( ( ( A e. P. /\ B e. P. ) /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
10	9	3ad2antl1 1143	. . . . . . 7 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ h e. Q. ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
11	10	ad2ant2r 500	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) )
12		prop 7283	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prarloc 7311	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
14	12, 13	sylan 281	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
15	14	adantll 467	. . . . . . . . . . . 12 |- ( ( ( A e. P. /\ B e. P. ) /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
16	15	3ad2antl1 1143	. . . . . . . . . . 11 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ h e. Q. ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
17	16	ad2ant2r 500	. . . . . . . . . 10 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
18	17	adantr 274	. . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) )
19		simpll1 1020	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( A e. P. /\ B e. P. ) )
20	19	ad2antrr 479	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( A e. P. /\ B e. P. ) )
21	20	simpld 111	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> A e. P. )
22	20	simprd 113	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> B e. P. )
23		simpll3 1022	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> q <Q r )
24	23	ad2antrr 479	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> q <Q r )
25		simplrl 524	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> h e. Q. )
26	25	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> h e. Q. )
27		simplrr 525	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q +Q p ) = r )
28		oveq2 5782	. . . . . . . . . . . . . . . 16 |- ( ( h +Q h ) = p -> ( q +Q ( h +Q h ) ) = ( q +Q p ) )
29	28	eqeq1d 2148	. . . . . . . . . . . . . . 15 |- ( ( h +Q h ) = p -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
30	29	ad2antll 482	. . . . . . . . . . . . . 14 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
31	27, 30	mpbird 166	. . . . . . . . . . . . 13 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q +Q ( h +Q h ) ) = r )
32	31	ad2antrr 479	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( q +Q ( h +Q h ) ) = r )
33		simprll 526	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> d e. ( 1st ` A ) )
34	33	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> d e. ( 1st ` A ) )
35		simprlr 527	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> u e. ( 2nd ` A ) )
36	35	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> u e. ( 2nd ` A ) )
37		simplrr 525	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> u <Q ( d +Q h ) )
38		simprll 526	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> e e. ( 1st ` B ) )
39		simprlr 527	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> t e. ( 2nd ` B ) )
40		simprr 521	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> t <Q ( e +Q h ) )
41	21, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40	addlocprlem 7343	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) /\ t <Q ( e +Q h ) ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
42	41	expr 372	. . . . . . . . . 10 |- ( ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) /\ ( e e. ( 1st ` B ) /\ t e. ( 2nd ` B ) ) ) -> ( t <Q ( e +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
43	42	rexlimdvva 2557	. . . . . . . . 9 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> ( E. e e. ( 1st ` B ) E. t e. ( 2nd ` B ) t <Q ( e +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
44	18, 43	mpd 13	. . . . . . . 8 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) /\ u <Q ( d +Q h ) ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
45	44	expr 372	. . . . . . 7 |- ( ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) /\ ( d e. ( 1st ` A ) /\ u e. ( 2nd ` A ) ) ) -> ( u <Q ( d +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
46	45	rexlimdvva 2557	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( E. d e. ( 1st ` A ) E. u e. ( 2nd ` A ) u <Q ( d +Q h ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
47	11, 46	mpd 13	. . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) /\ ( h e. Q. /\ ( h +Q h ) = p ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
48	5, 47	rexlimddv 2554	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) /\ ( p e. Q. /\ ( q +Q p ) = r ) ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
49	3, 48	rexlimddv 2554	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) )
50	49	3expia 1183	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( q e. Q. /\ r e. Q. ) ) -> ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )
51	50	ralrimivva 2514	1 |- ( ( A e. P. /\ B e. P. ) -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` ( A +P. B ) ) \/ r e. ( 2nd ` ( A +P. B ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2416   E.wrex 2417   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   +Q cplq 7090   <Q cltq 7093   P.cnp 7099   +P. cpp 7101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276

This theorem is referenced by:  addclpr  7345