Theorem vr1val 22209

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 7439	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
3	2	fveq1d 6909	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4		df-vr1 22198	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5		fvex 6920	. . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 7016	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅))
7	1, 6	eqtrid 2787	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8		fvprc 6899	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6951	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2800	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 21948	. . . . . 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 7567	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 7471	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
14	13	fveq1d 6909	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2778	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
16	7, 15	pm2.61i 182	1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  ∅c0 4339  ifcif 4531   ↦ cmpt 5231  ^◡ccnv 5688   “ cima 5692  ‘cfv 6563  (class class class)co 7431  1_oc1o 8498   ↑_m cmap 8865  Fincfn 8984  0cc0 11153  1c1 11154  ℕcn 12264  ℕ₀cn0 12524  0_gc0g 17486  1_rcur 20199   mVar cmvr 21943  var₁cv1 22193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mvr 21948  df-vr1 22198

This theorem is referenced by:  vr1cl2  22210  vr1cl  22235  subrgvr1  22280  subrgvr1cl  22281  coe1tm  22292  ply1coe  22318  evl1var  22356  evls1var  22358  rhmply1vr1  22407