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Theorem logbval 27759
 Description: Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 11967. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
Assertion
Ref Expression
logbval logb

Proof of Theorem logbval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5866 . . 3
21oveq2d 6301 . 2
3 fveq2 5866 . . 3
43oveq1d 6300 . 2
5 df-logb 27758 . 2 logb
6 ovex 6310 . 2
72, 4, 5, 6ovmpt2 6423 1 logb
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767   cdif 3473  csn 4027  cpr 4029  cfv 5588  (class class class)co 6285  cc 9491  cc0 9493  c1 9494   cdiv 10207  clog 22767  logbclogb 27757 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-logb 27758 This theorem is referenced by:  logb2aval  27760  logbcl  27764  logbid1  27765  rnlogbval  27767  relogbcl  27769  logb1  27770  nnlogbexp  27771  elogb  32467
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