Theorem vr1val 19383

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 6557	. . . . 5 ⊢ (𝑟 = 𝑅 → (1𝑜 mVar 𝑟) = (1𝑜 mVar 𝑅))
3	2	fveq1d 6105	. . . 4 ⊢ (𝑟 = 𝑅 → ((1𝑜 mVar 𝑟)‘∅) = ((1𝑜 mVar 𝑅)‘∅))
4		df-vr1 19372	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))
5		fvex 6113	. . . 4 ⊢ ((1𝑜 mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6191	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1𝑜 mVar 𝑅)‘∅))
7	1, 6	syl5eq 2656	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
8		fvprc 6097	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6137	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2669	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 19178	. . . . . 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	reldmmpt2 6669	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 6583	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1𝑜 mVar 𝑅) = ∅)
14	13	fveq1d 6105	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1𝑜 mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2647	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
16	7, 15	pm2.61i 175	1 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  ∅c0 3874  ifcif 4036   ↦ cmpt 4643  ^◡ccnv 5037   “ cima 5041  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440   ↑_𝑚 cmap 7744  Fincfn 7841  0cc0 9815  1c1 9816  ℕcn 10897  ℕ₀cn0 11169  0_gc0g 15923  1_rcur 18324   mVar cmvr 19173  var₁cv1 19367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-mvr 19178  df-vr1 19372

This theorem is referenced by:  vr1cl2  19384  vr1cl  19408  subrgvr1  19452  subrgvr1cl  19453  coe1tm  19464  ply1coe  19487  evl1var  19521  evls1var  19523