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Theorem axpre-mulgt0 9989
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 10112. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 10013. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
axpre-mulgt0 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
Proof of Theorem axpre-mulgt0
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elreal 9952 . 2 |- ( A e. RR <-> E. x e. R. <. x , 0R >. = A )
2 elreal 9952 . 2 |- ( B e. RR <-> E. y e. R. <. y , 0R >. = B )
3 breq2 4657 . . . 4 |- ( <. x , 0R >. = A -> ( 0 <RR <. x , 0R >. <-> 0 <RR A ) )
43anbi1d 741 . . 3 |- ( <. x , 0R >. = A -> ( ( 0 <RR <. x , 0R >. /\ 0 <RR <. y , 0R >. ) <-> ( 0 <RR A /\ 0 <RR <. y , 0R >. ) ) )
5 oveq1 6657 . . . 4 |- ( <. x , 0R >. = A -> ( <. x , 0R >. x. <. y , 0R >. ) = ( A x. <. y , 0R >. ) )
65breq2d 4665 . . 3 |- ( <. x , 0R >. = A -> ( 0 <RR ( <. x , 0R >. x. <. y , 0R >. ) <-> 0 <RR ( A x. <. y , 0R >. ) ) )
74, 6imbi12d 334 . 2 |- ( <. x , 0R >. = A -> ( ( ( 0 <RR <. x , 0R >. /\ 0 <RR <. y , 0R >. ) -> 0 <RR ( <. x , 0R >. x. <. y , 0R >. ) ) <-> ( ( 0 <RR A /\ 0 <RR <. y , 0R >. ) -> 0 <RR ( A x. <. y , 0R >. ) ) ) )
8 breq2 4657 . . . 4 |- ( <. y , 0R >. = B -> ( 0 <RR <. y , 0R >. <-> 0 <RR B ) )
98anbi2d 740 . . 3 |- ( <. y , 0R >. = B -> ( ( 0 <RR A /\ 0 <RR <. y , 0R >. ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
10 oveq2 6658 . . . 4 |- ( <. y , 0R >. = B -> ( A x. <. y , 0R >. ) = ( A x. B ) )
1110breq2d 4665 . . 3 |- ( <. y , 0R >. = B -> ( 0 <RR ( A x. <. y , 0R >. ) <-> 0 <RR ( A x. B ) ) )
129, 11imbi12d 334 . 2 |- ( <. y , 0R >. = B -> ( ( ( 0 <RR A /\ 0 <RR <. y , 0R >. ) -> 0 <RR ( A x. <. y , 0R >. ) ) <-> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) ) )
13 df-0 9943 . . . . . 6 |- 0 = <. 0R , 0R >.
1413breq1i 4660 . . . . 5 |- ( 0 <RR <. x , 0R >. <-> <. 0R , 0R >. <RR <. x , 0R >. )
15 ltresr 9961 . . . . 5 |- ( <. 0R , 0R >. <RR <. x , 0R >. <-> 0R <R x )
1614, 15bitri 264 . . . 4 |- ( 0 <RR <. x , 0R >. <-> 0R <R x )
1713breq1i 4660 . . . . 5 |- ( 0 <RR <. y , 0R >. <-> <. 0R , 0R >. <RR <. y , 0R >. )
18 ltresr 9961 . . . . 5 |- ( <. 0R , 0R >. <RR <. y , 0R >. <-> 0R <R y )
1917, 18bitri 264 . . . 4 |- ( 0 <RR <. y , 0R >. <-> 0R <R y )
20 mulgt0sr 9926 . . . 4 |- ( ( 0R <R x /\ 0R <R y ) -> 0R <R ( x .R y ) )
2116, 19, 20syl2anb 496 . . 3 |- ( ( 0 <RR <. x , 0R >. /\ 0 <RR <. y , 0R >. ) -> 0R <R ( x .R y ) )
2213a1i 11 . . . . 5 |- ( ( x e. R. /\ y e. R. ) -> 0 = <. 0R , 0R >. )
23 mulresr 9960 . . . . 5 |- ( ( x e. R. /\ y e. R. ) -> ( <. x , 0R >. x. <. y , 0R >. ) = <. ( x .R y ) , 0R >. )
2422, 23breq12d 4666 . . . 4 |- ( ( x e. R. /\ y e. R. ) -> ( 0 <RR ( <. x , 0R >. x. <. y , 0R >. ) <-> <. 0R , 0R >. <RR <. ( x .R y ) , 0R >. ) )
25 ltresr 9961 . . . 4 |- ( <. 0R , 0R >. <RR <. ( x .R y ) , 0R >. <-> 0R <R ( x .R y ) )
2624, 25syl6bb 276 . . 3 |- ( ( x e. R. /\ y e. R. ) -> ( 0 <RR ( <. x , 0R >. x. <. y , 0R >. ) <-> 0R <R ( x .R y ) ) )
2721, 26syl5ibr 236 . 2 |- ( ( x e. R. /\ y e. R. ) -> ( ( 0 <RR <. x , 0R >. /\ 0 <RR <. y , 0R >. ) -> 0 <RR ( <. x , 0R >. x. <. y , 0R >. ) ) )
281, 2, 7, 12, 272gencl 3236 1 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653  (class class class)co 6650   R.cnr 9687   0Rc0r 9688   .R cmr 9692   <R cltr 9693   RRcr 9935   0cc0 9936   <RR cltrr 9940   x. cmul 9941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-1p 9804  df-plp 9805  df-mp 9806  df-ltp 9807  df-enr 9877  df-nr 9878  df-plr 9879  df-mr 9880  df-ltr 9881  df-0r 9882  df-m1r 9884  df-c 9942  df-0 9943  df-r 9946  df-mul 9948  df-lt 9949
This theorem is referenced by: (None)
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