Theorem axarch 8208

Description: Archimedean axiom. The Archimedean property is more naturally stated once we have defined NN. Unless we find another way to state it, we'll just use the right hand side of dfnn2 9241 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 8248. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axarch	|- ( A e. RR -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )

Distinct variable group:   A, n, x, y

Proof of Theorem axarch

Dummy variables l u w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 8145	. . 3 |- ( A e. RR <-> E. z e. R. <. z , 0R >. = A )
2	1	biimpi 120	. 2 |- ( A e. RR -> E. z e. R. <. z , 0R >. = A )
3		archsr 8099	. . . 4 |- ( z e. R. -> E. w e. N. z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
4	3	ad2antrl 490	. . 3 |- ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) -> E. w e. N. z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
5		simplrr 538	. . . . 5 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> <. z , 0R >. = A )
6		simprr 533	. . . . . 6 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
7		ltresr 8156	. . . . . 6 |- ( <. z , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <-> z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
8	6, 7	sylibr 134	. . . . 5 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> <. z , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
9	5, 8	eqbrtrrd 4135	. . . 4 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> A <RR <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
10		pitonn 8165	. . . . . 6 |- ( w e. N. -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
11	10	ad2antrl 490	. . . . 5 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
12		simpr 110	. . . . . 6 |- ( ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) /\ n = <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) -> n = <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
13	12	breq2d 4123	. . . . 5 |- ( ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) /\ n = <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) -> ( A <RR n <-> A <RR <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
14	11, 13	rspcedv 2927	. . . 4 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> ( A <RR <. [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n ) )
15	9, 14	mpd 13	. . 3 |- ( ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) /\ ( w e. N. /\ z <R [ <. ( <. { l | l <Q [ <. w , 1o >. ] ~Q } , { u | [ <. w , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )
16	4, 15	rexlimddv 2667	. 2 |- ( ( A e. RR /\ ( z e. R. /\ <. z , 0R >. = A ) ) -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )
17	2, 16	rexlimddv 2667	1 |- ( A e. RR -> E. n e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } A <RR n )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   <.cop 3694   |^|cint 3951   class class class wbr 4111  (class class class)co 6052   1oc1o 6642   [cec 6767   N.cnpi 7589   ~Q ceq 7596   <Q cltq 7602   1Pc1p 7609   +P. cpp 7610   ~R cer 7613   R.cnr 7614   0Rc0r 7615   <R cltr 7620   RRcr 8128   1c1 8130   + caddc 8132   <RR cltrr 8133

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-2o 6650  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7621  df-pli 7622  df-mi 7623  df-lti 7624  df-plpq 7661  df-mpq 7662  df-enq 7664  df-nqqs 7665  df-plqqs 7666  df-mqqs 7667  df-1nqqs 7668  df-rq 7669  df-ltnqqs 7670  df-enq0 7741  df-nq0 7742  df-0nq0 7743  df-plq0 7744  df-mq0 7745  df-inp 7783  df-i1p 7784  df-iplp 7785  df-iltp 7787  df-enr 8043  df-nr 8044  df-plr 8045  df-ltr 8047  df-0r 8048  df-1r 8049  df-c 8135  df-1 8137  df-r 8139  df-add 8140  df-lt 8142

This theorem is referenced by: (None)