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Theorem deg1fval 25468
Description: Relate univariate polynomial degree to multivariate. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypothesis
Ref Expression
deg1fval.d 𝐷 = ( deg1𝑅)
Assertion
Ref Expression
deg1fval 𝐷 = (1o mDeg 𝑅)

Proof of Theorem deg1fval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 deg1fval.d . 2 𝐷 = ( deg1𝑅)
2 oveq2 7369 . . . 4 (𝑟 = 𝑅 → (1o mDeg 𝑟) = (1o mDeg 𝑅))
3 df-deg1 25441 . . . 4 deg1 = (𝑟 ∈ V ↦ (1o mDeg 𝑟))
4 ovex 7394 . . . 4 (1o mDeg 𝑅) ∈ V
52, 3, 4fvmpt 6952 . . 3 (𝑅 ∈ V → ( deg1𝑅) = (1o mDeg 𝑅))
6 fvprc 6838 . . . 4 𝑅 ∈ V → ( deg1𝑅) = ∅)
7 reldmmdeg 25442 . . . . 5 Rel dom mDeg
87ovprc2 7401 . . . 4 𝑅 ∈ V → (1o mDeg 𝑅) = ∅)
96, 8eqtr4d 2776 . . 3 𝑅 ∈ V → ( deg1𝑅) = (1o mDeg 𝑅))
105, 9pm2.61i 182 . 2 ( deg1𝑅) = (1o mDeg 𝑅)
111, 10eqtri 2761 1 𝐷 = (1o mDeg 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  cfv 6500  (class class class)co 7361  1oc1o 8409   mDeg cmdg 25438   deg1 cdg1 25439
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-mdeg 25440  df-deg1 25441
This theorem is referenced by:  deg1xrf  25469  deg1cl  25471  deg1propd  25474  deg1z  25475  deg1nn0cl  25476  deg1ldg  25480  deg1leb  25483  deg1val  25484  deg1addle  25489  deg1vscale  25492  deg1vsca  25493  deg1mulle2  25497  deg1le0  25499
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