Theorem addlocpr 7477

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 7444 to both A and B, and uses nqtri3or 7337 rather than prloc 7432 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 7349	. . . . . 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 1005	. . . 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 7351	. . . . . 6 |- ( p e. Q. -> E. h e. Q. ( h +Q h ) = p )
5	4	ad2antrl 482	. . . . 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 7416	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7444	. . . . . . . . . 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 469	. . . . . . . 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 1149	. . . . . . 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 501	. . . . . 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 7416	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prarloc 7444	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . 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 1149	. . . . . . . . . . 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 501	. . . . . . . . . 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 1026	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 1028	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 525	. . . . . . . . . . . . 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 526	. . . . . . . . . . . . . 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 5850	. . . . . . . . . . . . . . . 16 |- ( ( h +Q h ) = p -> ( q +Q ( h +Q h ) ) = ( q +Q p ) )
29	28	eqeq1d 2174	. . . . . . . . . . . . . . 15 |- ( ( h +Q h ) = p -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
30	29	ad2antll 483	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 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 ) ) ) -> 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 528	. . . . . . . . . . . . 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 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 ) ) ) -> u <Q ( d +Q h ) )
38		simprll 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 ) ) ) -> e e. ( 1st ` B ) )
39		simprlr 528	. . . . . . . . . . . 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 522	. . . . . . . . . . . 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 7476	. . . . . . . . . . 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 373	. . . . . . . . . 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 2591	. . . . . . . . 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 373	. . . . . . 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 2591	. . . . . 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 2588	. . . 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 2588	. . 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 1195	. 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 2548	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 698   /\ w3a 968   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409

This theorem is referenced by:  addclpr  7478