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Theorem mulgt0sr 6684
Description: The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
Assertion
Ref Expression
mulgt0sr  0R  <R  0R  <R 
0R  <R  .R

Proof of Theorem mulgt0sr
Dummy variables  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelsr 6646 . . . . 5  <R  C_  R.  X.  R.
21brel 4335 . . . 4  0R 
<R  0R  R.  R.
32simprd 107 . . 3  0R 
<R  R.
41brel 4335 . . . 4  0R 
<R  0R  R.  R.
54simprd 107 . . 3  0R 
<R  R.
63, 5anim12i 321 . 2  0R  <R  0R  <R  R.  R.
7 df-nr 6635 . . 3  R.  P.  X.  P. /.  ~R
8 breq2 3759 . . . . 5  <. ,  >. 
~R  0R  <R  <. ,  >.  ~R  0R  <R
98anbi1d 438 . . . 4  <. ,  >. 
~R  0R  <R  <. ,  >.  ~R  0R  <R  <. ,  >.  ~R  0R  <R  0R  <R  <. ,  >.  ~R
10 oveq1 5462 . . . . 5  <. ,  >. 
~R  <. , 
>.  ~R  .R  <. ,  >. 
~R  .R  <. ,  >.  ~R
1110breq2d 3767 . . . 4  <. ,  >. 
~R  0R  <R  <. ,  >. 
~R  .R  <. ,  >.  ~R  0R  <R  .R 
<. ,  >. 
~R
129, 11imbi12d 223 . . 3  <. ,  >. 
~R  0R  <R 
<. ,  >. 
~R  0R  <R 
<. ,  >. 
~R  0R  <R  <. ,  >.  ~R  .R 
<. ,  >. 
~R  0R  <R  0R  <R  <. ,  >.  ~R  0R  <R  .R  <. ,  >.  ~R
13 breq2 3759 . . . . 5  <. ,  >. 
~R  0R  <R  <. ,  >.  ~R  0R  <R
1413anbi2d 437 . . . 4  <. ,  >. 
~R  0R  <R  0R  <R  <. ,  >.  ~R  0R  <R  0R  <R
15 oveq2 5463 . . . . 5  <. ,  >. 
~R  .R  <. ,  >.  ~R  .R
1615breq2d 3767 . . . 4  <. ,  >. 
~R  0R  <R 
.R  <. ,  >.  ~R  0R  <R  .R
1714, 16imbi12d 223 . . 3  <. ,  >. 
~R  0R  <R  0R  <R  <. ,  >.  ~R  0R  <R  .R  <. ,  >.  ~R  0R  <R  0R  <R  0R  <R  .R
18 gt0srpr 6656 . . . . 5  0R 
<R  <. , 
>.  ~R  <P
19 gt0srpr 6656 . . . . 5  0R 
<R  <. ,  >.  ~R  <P
2018, 19anbi12i 433 . . . 4  0R  <R  <. ,  >.  ~R  0R  <R  <. ,  >.  ~R  <P  <P
21 ltexpri 6586 . . . . . . 7 
<P  P.  +P.
22 ltexpri 6586 . . . . . . . . 9 
<P  P.  +P.
23 addclpr 6519 . . . . . . . . . . . . . 14  P.  P.  +P.  P.
2423adantl 262 . . . . . . . . . . . . 13  P.  P.  P.  P. 
P.  +P.  P.  +P.  P.  P.  +P.  P.
25 simplrr 488 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P.
26 simplr 482 . . . . . . . . . . . . . . . . 17  P.  P.  P.  P.  P.
2726ad2antrr 457 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
28 simplrl 487 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
2924, 27, 28caovcld 5596 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
P.
3025, 29eqeltrrd 2112 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
31 simplrr 488 . . . . . . . . . . . . . . 15 
P.  P.  P.  P.  P.  +P.  P.
3231adantr 261 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
33 mulclpr 6552 . . . . . . . . . . . . . 14  P.  P.  .P.  P.
3430, 32, 33syl2anc 391 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
35 simplrl 487 . . . . . . . . . . . . . . 15 
P.  P.  P.  P.  P.  +P.  P.
3635adantr 261 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
37 mulclpr 6552 . . . . . . . . . . . . . 14  P.  P.  .P.  P.
3827, 36, 37syl2anc 391 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
3924, 34, 38caovcld 5596 . . . . . . . . . . . 12  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  P.
40 simprl 483 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  P.
41 mulclpr 6552 . . . . . . . . . . . . 13  P.  P.  .P.  P.
4228, 40, 41syl2anc 391 . . . . . . . . . . . 12  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
43 ltaddpr 6570 . . . . . . . . . . . 12  .P.  +P. 
.P.  P.  .P. 
P.  .P. 
+P.  .P.  <P  .P.  +P.  .P.  +P.  .P.
4439, 42, 43syl2anc 391 . . . . . . . . . . 11  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  <P  .P.  +P.  .P.  +P.  .P.
45 simprr 484 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P.
46 oveq12 5464 . . . . . . . . . . . . . . . 16  +P.  +P.  +P.  .P.  +P.  .P.
4746oveq1d 5470 . . . . . . . . . . . . . . 15  +P.  +P.  +P.  .P. 
+P. 
+P. 
.P.  +P.  .P.  .P.  +P. 
.P.  +P.  .P.
4825, 45, 47syl2anc 391 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P.  .P.  +P.  +P.  .P. 
+P.  .P.  .P.  +P.  .P. 
+P.  .P.
49 distrprg 6563 . . . . . . . . . . . . . . . . . . 19  P.  P.  P.  .P.  +P.  .P.  +P.  .P.
5027, 32, 40, 49syl3anc 1134 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P.
51 oveq2 5463 . . . . . . . . . . . . . . . . . . . 20  +P.  .P.  +P.  .P.
5251adantl 262 . . . . . . . . . . . . . . . . . . 19  P.  +P.  .P. 
+P.  .P.
5352adantl 262 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.
5450, 53eqtr3d 2071 . . . . . . . . . . . . . . . . 17  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  .P.
5554oveq1d 5470 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
+P.  .P.  .P.  +P.  .P. 
+P.  .P.
56 distrprg 6563 . . . . . . . . . . . . . . . . . . . 20  P.  P.  h  P.  .P.  +P.  h  .P.  +P. 
.P.  h
5756adantl 262 . . . . . . . . . . . . . . . . . . 19  P.  P.  P.  P. 
P.  +P.  P.  +P.  P.  P.  h  P.  .P.  +P.  h  .P. 
+P.  .P.  h
58 mulcomprg 6555 . . . . . . . . . . . . . . . . . . . 20  P.  P.  .P.  .P.
5958adantl 262 . . . . . . . . . . . . . . . . . . 19  P.  P.  P.  P. 
P.  +P.  P.  +P.  P.  P.  .P.  .P.
6057, 27, 28, 32, 24, 59caovdir2d 5619 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  .P.  +P. 
.P.
6157, 27, 28, 40, 24, 59caovdir2d 5619 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  .P.  +P. 
.P.
6260, 61oveq12d 5473 . . . . . . . . . . . . . . . . 17  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P.  .P.  +P.  +P.  .P.  .P. 
+P.  .P.  +P.  .P. 
+P.  .P.
63 distrprg 6563 . . . . . . . . . . . . . . . . . 18  +P.  P.  P. 
P.  +P. 
.P.  +P.  +P.  .P. 
+P. 
+P.  .P.
6429, 32, 40, 63syl3anc 1134 . . . . . . . . . . . . . . . . 17  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  +P.  +P.  .P. 
+P. 
+P.  .P.
65 mulclpr 6552 . . . . . . . . . . . . . . . . . . 19  P.  P.  .P.  P.
6627, 32, 65syl2anc 391 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
67 mulclpr 6552 . . . . . . . . . . . . . . . . . . 19  P.  P.  .P.  P.
6827, 40, 67syl2anc 391 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
69 mulclpr 6552 . . . . . . . . . . . . . . . . . . 19  P.  P.  .P.  P.
7028, 32, 69syl2anc 391 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
71 addcomprg 6553 . . . . . . . . . . . . . . . . . . 19  P.  P.  +P.  +P.
7271adantl 262 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P. 
P.  +P.  P.  +P.  P.  P.  +P.  +P.
73 addassprg 6554 . . . . . . . . . . . . . . . . . . 19  P.  P.  h  P.  +P. 
+P.  h  +P.  +P.  h
7473adantl 262 . . . . . . . . . . . . . . . . . 18  P.  P.  P.  P. 
P.  +P.  P.  +P.  P.  P.  h  P. 
+P.  +P.  h  +P.  +P.  h
7566, 68, 70, 72, 74, 42, 24caov4d 5627 . . . . . . . . . . . . . . . . 17  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
+P.  .P. 
.P.  +P.  .P.  +P.  .P. 
+P.  .P.
7662, 64, 753eqtr4d 2079 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  +P.  .P.  +P. 
.P. 
+P. 
.P.  +P.  .P.
7770, 38, 42, 72, 74caov12d 5624 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P. 
.P.  +P.  .P.  .P.  +P. 
.P.  +P.  .P.
7855, 76, 773eqtr4d 2079 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  +P.  .P.  +P. 
.P.  +P.  .P.
79 oveq1 5462 . . . . . . . . . . . . . . . . . 18  +P.  +P. 
.P.  .P.
8079adantl 262 . . . . . . . . . . . . . . . . 17  P.  +P.  +P.  .P.  .P.
8180ad2antlr 458 . . . . . . . . . . . . . . . 16  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P. 
.P.  .P.
8260, 81eqtr3d 2071 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  .P.
8378, 82oveq12d 5473 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  +P.  .P.  +P.  +P.  .P. 
+P.  .P. 
.P.  +P.  .P.  +P.  .P.  +P.  .P.
8448, 83eqtr3d 2071 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P. 
.P.  +P.  .P.  .P.  +P.  .P. 
+P.  .P.  +P.  .P.
85 mulclpr 6552 . . . . . . . . . . . . . . . 16  P.  P.  .P.  P.
8630, 36, 85syl2anc 391 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
P.
87 addassprg 6554 . . . . . . . . . . . . . . 15  .P.  P.  .P. 
P.  .P.  P.  .P. 
+P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
8886, 66, 70, 87syl3anc 1134 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
89 addclpr 6519 . . . . . . . . . . . . . . . 16  .P.  P.  .P. 
P.  .P. 
+P.  .P.  P.
9086, 66, 89syl2anc 391 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  P.
91 addcomprg 6553 . . . . . . . . . . . . . . 15  .P.  +P. 
.P.  P.  .P. 
P.  .P.  +P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
9290, 70, 91syl2anc 391 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
9388, 92eqtr3d 2071 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P. 
.P.  +P.  .P.  .P.  +P. 
.P.  +P.  .P.
9424, 38, 42caovcld 5596 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  P.
95 addassprg 6554 . . . . . . . . . . . . . . 15  .P.  P.  .P. 
P.  .P.  +P.  .P.  P. 
.P.  +P.  .P.  +P.  .P. 
+P.  .P.  .P.  +P.  .P. 
+P. 
.P.  +P.  .P.
9670, 34, 94, 95syl3anc 1134 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
+P.  .P.  .P.  +P.  .P. 
+P. 
.P.  +P.  .P.
9770, 94, 34, 72, 74caov32d 5623 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P. 
.P.  +P.  .P.  +P.  .P.  .P. 
+P.  .P.  +P.  .P. 
+P.  .P.
98 addassprg 6554 . . . . . . . . . . . . . . . 16  .P.  P.  .P. 
P.  .P.  P.  .P. 
+P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
9934, 38, 42, 98syl3anc 1134 . . . . . . . . . . . . . . 15  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P.  .P.
10099oveq2d 5471 . . . . . . . . . . . . . 14  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P. 
+P.  .P.  +P.  .P.  .P. 
+P. 
.P.  +P.  .P.  +P.  .P.
10196, 97, 1003eqtr4d 2079 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P. 
.P.  +P.  .P.  +P.  .P. 
.P.  +P.  .P.  +P. 
.P. 
+P.  .P.
10284, 93, 1013eqtr3d 2077 . . . . . . . . . . . 12  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P. 
.P.  +P.  .P.  .P.  +P.  .P. 
+P.  .P.  +P.  .P.
10324, 39, 42caovcld 5596 . . . . . . . . . . . . 13  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P.  .P.  +P.  .P.  P.
104 addcanprg 6589 . . . . . . . . . . . . 13  .P.  P.  .P. 
+P.  .P.  P. 
.P.  +P.  .P.  +P. 
.P.  P.  .P.  +P.  .P. 
+P.  .P.  .P.  +P.  .P.  +P.  .P.  +P.  .P.  .P. 
+P.  .P.  .P.  +P. 
.P. 
+P.  .P.
10570, 90, 103, 104syl3anc 1134 . . . . . . . . . . . 12  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P.  +P. 
.P.  +P.  .P.  .P.  +P.  .P. 
+P.  .P.  +P.  .P.  .P. 
+P.  .P.  .P.  +P. 
.P. 
+P.  .P.
106102, 105mpd 13 . . . . . . . . . . 11  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  .P.  +P. 
.P. 
+P.  .P.
10744, 106breqtrrd 3781 . . . . . . . . . 10  P.  P.  P.  P.  P.  +P. 
P.  +P.  .P. 
+P.  .P.  <P  .P. 
+P.  .P.
108107rexlimdvaa 2428 . . . . . . . . 9 
P.  P.  P.  P.  P.  +P.  P.  +P.  .P.  +P. 
.P. 
<P  .P.  +P. 
.P.
10922, 108syl5 28 . . . . . . . 8 
P.  P.  P.  P.  P.  +P.  <P  .P.  +P. 
.P. 
<P  .P.  +P. 
.P.
110109rexlimdvaa 2428 . . . . . . 7  P.  P.  P.  P.  P.  +P.  <P  .P.  +P. 
.P. 
<P  .P.  +P. 
.P.
11121, 110syl5 28 . . . . . 6  P.  P.  P.  P.  <P  <P  .P.  +P. 
.P. 
<P  .P.  +P. 
.P.
112111impd 242 . . . . 5  P.  P.  P.  P. 
<P  <P  .P. 
+P.  .P.  <P  .P. 
+P.  .P.
113 mulsrpr 6654 . . . . . . 7  P.  P.  P.  P.  <. ,  >.  ~R  .R  <. ,  >.  ~R  <. 
.P.  +P.  .P.  ,  .P. 
+P.  .P.  >.  ~R
114113breq2d 3767 . . . . . 6  P.  P.  P.  P.  0R  <R  <. , 
>.  ~R  .R  <. ,  >. 
~R  0R  <R 
<.  .P.  +P. 
.P.  , 
.P.  +P.  .P.  >.  ~R
115 gt0srpr 6656 . . . . . 6  0R 
<R  <. 
.P.  +P.  .P.  ,  .P. 
+P.  .P.  >.  ~R  .P.  +P.  .P.  <P  .P. 
+P.  .P.
116114, 115syl6bb 185 . . . . 5  P.  P.  P.  P.  0R  <R  <. , 
>.  ~R  .R  <. ,  >. 
~R  .P. 
+P.  .P.  <P  .P. 
+P.  .P.
117112, 116sylibrd 158 . . . 4  P.  P.  P.  P. 
<P  <P  0R  <R  <. ,  >.  ~R  .R 
<. ,  >. 
~R
11820, 117syl5bi 141 . . 3  P.  P.  P.  P.  0R 
<R  <. , 
>.  ~R  0R  <R  <. ,  >.  ~R 
0R  <R  <. ,  >.  ~R  .R  <. ,  >.  ~R
1197, 12, 17, 1182ecoptocl 6130 . 2  R.  R.  0R  <R  0R  <R  0R  <R  .R
1206, 119mpcom 32 1  0R  <R  0R  <R 
0R  <R  .R
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   w3a 884   wceq 1242   wcel 1390  wrex 2301   <.cop 3370   class class class wbr 3755  (class class class)co 5455  cec 6040   P.cnp 6275    +P. cpp 6277    .P. cmp 6278    <P cltp 6279    ~R cer 6280   R.cnr 6281   0Rc0r 6282    .R cmr 6286    <R cltr 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-i1p 6449  df-iplp 6450  df-imp 6451  df-iltp 6452  df-enr 6634  df-nr 6635  df-mr 6637  df-ltr 6638  df-0r 6639
This theorem is referenced by:  axpre-mulgt0  6751
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