Theorem logbval 26271

Description: Define the value of the log_b 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 argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)

Assertion

Ref	Expression
logbval	⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Proof of Theorem logbval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6892	. . 3 ⊢ (𝑥 = 𝐵 → (log‘𝑥) = (log‘𝐵))
2	1	oveq2d 7425	. 2 ⊢ (𝑥 = 𝐵 → ((log‘𝑦) / (log‘𝑥)) = ((log‘𝑦) / (log‘𝐵)))
3		fveq2 6892	. . 3 ⊢ (𝑦 = 𝑋 → (log‘𝑦) = (log‘𝑋))
4	3	oveq1d 7424	. 2 ⊢ (𝑦 = 𝑋 → ((log‘𝑦) / (log‘𝐵)) = ((log‘𝑋) / (log‘𝐵)))
5		df-logb 26270	. 2 ⊢ logb = (𝑥 ∈ (ℂ ∖ {0, 1}), 𝑦 ∈ (ℂ ∖ {0}) ↦ ((log‘𝑦) / (log‘𝑥)))
6		ovex 7442	. 2 ⊢ ((log‘𝑋) / (log‘𝐵)) ∈ V
7	2, 4, 5, 6	ovmpo 7568	1 ⊢ ((𝐵 ∈ (ℂ ∖ {0, 1}) ∧ 𝑋 ∈ (ℂ ∖ {0})) → (𝐵 logb 𝑋) = ((log‘𝑋) / (log‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ∖ cdif 3946  {csn 4629  {cpr 4631  ‘cfv 6544  (class class class)co 7409  ℂcc 11108  0cc0 11110  1c1 11111   / cdiv 11871  logclog 26063   log_b clogb 26269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-logb 26270

This theorem is referenced by:  logbcl  26272  logbid1  26273  logb1  26274  elogb  26275  logbchbase  26276  relogbval  26277  relogbcl  26278  relogbreexp  26280  relogbmul  26282  nnlogbexp  26286  relogbcxp  26290  cxplogb  26291  logbgt0b  26298  dvrelog2b  40931  dvrelogpow2b  40933  rege1logbrege0  47244  logb2aval  47809