Theorem vr1val 22151

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 7427	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
3	2	fveq1d 6898	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4		df-vr1 22140	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5		fvex 6909	. . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 7004	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅))
7	1, 6	eqtrid 2777	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8		fvprc 6888	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6940	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2790	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 21877	. . . . . 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 7555	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 7459	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
14	13	fveq1d 6898	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2768	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
16	7, 15	pm2.61i 182	1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  {crab 3418  Vcvv 3461  ∅c0 4322  ifcif 4530   ↦ cmpt 5232  ^◡ccnv 5677   “ cima 5681  ‘cfv 6549  (class class class)co 7419  1_oc1o 8480   ↑_m cmap 8845  Fincfn 8964  0cc0 11145  1c1 11146  ℕcn 12250  ℕ₀cn0 12510  0_gc0g 17440  1_rcur 20150   mVar cmvr 21872  var₁cv1 22135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-mvr 21877  df-vr1 22140

This theorem is referenced by:  vr1cl2  22152  vr1cl  22177  subrgvr1  22222  subrgvr1cl  22223  coe1tm  22234  ply1coe  22259  evl1var  22297  evls1var  22299  rhmply1vr1  22348