Theorem vr1val 22052

Description: The value of the generator of the power series algebra (the 𝑋 in 𝑅[[𝑋]]). Since all univariate polynomial rings over a fixed base ring 𝑅 are isomorphic, we don't bother to pass this in as a parameter; internally we are actually using the empty set as this generator and 1_o = {∅} is the index set (but for most purposes this choice should not be visible anyway). (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)

Hypothesis

Ref	Expression
vr1val.1	⊢ 𝑋 = (var1‘𝑅)

Assertion

Ref	Expression
vr1val	⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Proof of Theorem vr1val

Dummy variables 𝑓 ℎ 𝑖 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vr1val.1	. . 3 ⊢ 𝑋 = (var1‘𝑅)
2		oveq2 7377	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
3	2	fveq1d 6842	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4		df-vr1 22041	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5		fvex 6853	. . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6950	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅))
7	1, 6	eqtrid 2776	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8		fvprc 6832	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6884	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2789	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 21795	. . . . . 6 ⊢ mVar = (𝑖 ∈ V, 𝑟 ∈ V ↦ (𝑥 ∈ 𝑖 ↦ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 𝑖) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 𝑖 ↦ if(𝑦 = 𝑥, 1, 0)), (1r‘𝑟), (0g‘𝑟)))))
12	11	reldmmpo 7503	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 7409	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
14	13	fveq1d 6842	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2767	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
16	7, 15	pm2.61i 182	1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444  ∅c0 4292  ifcif 4484   ↦ cmpt 5183  ^◡ccnv 5630   “ cima 5634  ‘cfv 6499  (class class class)co 7369  1_oc1o 8404   ↑_m cmap 8776  Fincfn 8895  0cc0 11044  1c1 11045  ℕcn 12162  ℕ₀cn0 12418  0_gc0g 17378  1_rcur 20066   mVar cmvr 21790  var₁cv1 22036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-mvr 21795  df-vr1 22041

This theorem is referenced by:  vr1cl2  22053  vr1cl  22078  subrgvr1  22123  subrgvr1cl  22124  coe1tm  22135  ply1coe  22161  evl1var  22199  evls1var  22201  rhmply1vr1  22250