Theorem addlocpr 7498

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 7465 to both A and B, and uses nqtri3or 7358 rather than prloc 7453 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 7370	. . . . . 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 1010	. . . 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 7372	. . . . . 6 |- ( p e. Q. -> E. h e. Q. ( h +Q h ) = p )
5	4	ad2antrl 487	. . . . 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 7437	. . . . . . . . . 10 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7465	. . . . . . . . . 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 474	. . . . . . . 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 1154	. . . . . . 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 506	. . . . . 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 7437	. . . . . . . . . . . . . 14 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
13		prarloc 7465	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . 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 1154	. . . . . . . . . . 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 506	. . . . . . . . . 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 1031	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 1033	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 530	. . . . . . . . . . . . 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 531	. . . . . . . . . . . . . 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 5861	. . . . . . . . . . . . . . . 16 |- ( ( h +Q h ) = p -> ( q +Q ( h +Q h ) ) = ( q +Q p ) )
29	28	eqeq1d 2179	. . . . . . . . . . . . . . 15 |- ( ( h +Q h ) = p -> ( ( q +Q ( h +Q h ) ) = r <-> ( q +Q p ) = r ) )
30	29	ad2antll 488	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 532	. . . . . . . . . . . . 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 533	. . . . . . . . . . . . 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 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 ) ) ) -> u <Q ( d +Q h ) )
38		simprll 532	. . . . . . . . . . . 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 533	. . . . . . . . . . . 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 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 <Q ( e +Q h ) )
41	21, 22, 24, 26, 32, 34, 36, 37, 38, 39, 40	addlocprlem 7497	. . . . . . . . . . 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 2595	. . . . . . . . 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 2595	. . . . . 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 2592	. . . 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 2592	. . 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 1200	. 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 2552	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 703   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   +Q cplq 7244   <Q cltq 7247   P.cnp 7253   +P. cpp 7255

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430

This theorem is referenced by:  addclpr  7499