Theorem vr1val 19562

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 6658	. . . . 5 ⊢ (𝑟 = 𝑅 → (1𝑜 mVar 𝑟) = (1𝑜 mVar 𝑅))
3	2	fveq1d 6193	. . . 4 ⊢ (𝑟 = 𝑅 → ((1𝑜 mVar 𝑟)‘∅) = ((1𝑜 mVar 𝑅)‘∅))
4		df-vr1 19551	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))
5		fvex 6201	. . . 4 ⊢ ((1𝑜 mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6282	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1𝑜 mVar 𝑅)‘∅))
7	1, 6	syl5eq 2668	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
8		fvprc 6185	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6227	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2681	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 19357	. . . . . 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 6771	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 6685	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1𝑜 mVar 𝑅) = ∅)
14	13	fveq1d 6193	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1𝑜 mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2659	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
16	7, 15	pm2.61i 176	1 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  ∅c0 3915  ifcif 4086   ↦ cmpt 4729  ^◡ccnv 5113   “ cima 5117  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937  ℕcn 11020  ℕ₀cn0 11292  0_gc0g 16100  1_rcur 18501   mVar cmvr 19352  var₁cv1 19546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mvr 19357  df-vr1 19551

This theorem is referenced by:  vr1cl2  19563  vr1cl  19587  subrgvr1  19631  subrgvr1cl  19632  coe1tm  19643  ply1coe  19666  evl1var  19700  evls1var  19702