Theorem addlocpr 7620

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 7587 to both A and B, and uses nqtri3or 7480 rather than prloc 7575 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 7492	. . . . . 6 |- ( ( q e. Q. /\ r e. Q. ) -> ( q <Q r <-> E. p e. Q. ( q +Q p ) = r ) )
2	1	biimpa 296	. . . . 5 |- ( ( ( q e. Q. /\ r e. Q. ) /\ q <Q r ) -> E. p e. Q. ( q +Q p ) = r )
3	2	3adant1 1017	. . . 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 7494	. . . . . 6 |- ( p e. Q. -> E. h e. Q. ( h +Q h ) = p )
5	4	ad2antrl 490	. . . . 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 7559	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7587	. . . . . . . . . 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 283	. . . . . . . . 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 477	. . . . . . . 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 1161	. . . . . . 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 509	. . . . . 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 7559	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prarloc 7587	. . . . . . . . . . . . . 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 283	. . . . . . . . . . . . 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 476	. . . . . . . . . . . 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 1161	. . . . . . . . . . 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 509	. . . . . . . . . 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 276	. . . . . . . . 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 1038	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 112	. . . . . . . . . . . 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 114	. . . . . . . . . . . 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 1040	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 535	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 536	. . . . . . . . . . . . . 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 5933	. . . . . . . . . . . . . . . 16 |- ( ( h +Q h ) = p -> ( q +Q ( h +Q h ) ) = ( q +Q p ) )
29	28	eqeq1d 2205	. . . . . . . . . . . . . . 15 |- ( ( h +Q h ) = p -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
30	29	ad2antll 491	. . . . . . . . . . . . . 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 167	. . . . . . . . . . . . 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 488	. . . . . . . . . . . 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 537	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 538	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 536	. . . . . . . . . . . 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 537	. . . . . . . . . . . 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 538	. . . . . . . . . . . 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 531	. . . . . . . . . . . 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 7619	. . . . . . . . . . 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 375	. . . . . . . . . 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 2622	. . . . . . . . 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 375	. . . . . . 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 2622	. . . . . 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 2619	. . . 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 2619	. . 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 1207	. 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 2579	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 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   A.wral 2475   E.wrex 2476   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   P.cnp 7375   +P. cpp 7377

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552

This theorem is referenced by:  addclpr  7621