Theorem vr1val 19776

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 6800	. . . . 5 ⊢ (𝑟 = 𝑅 → (1𝑜 mVar 𝑟) = (1𝑜 mVar 𝑅))
3	2	fveq1d 6334	. . . 4 ⊢ (𝑟 = 𝑅 → ((1𝑜 mVar 𝑟)‘∅) = ((1𝑜 mVar 𝑅)‘∅))
4		df-vr1 19765	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1𝑜 mVar 𝑟)‘∅))
5		fvex 6342	. . . 4 ⊢ ((1𝑜 mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6424	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1𝑜 mVar 𝑅)‘∅))
7	1, 6	syl5eq 2816	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
8		fvprc 6326	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6368	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2829	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 19571	. . . . . 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 6917	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 6829	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1𝑜 mVar 𝑅) = ∅)
14	13	fveq1d 6334	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1𝑜 mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2807	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1𝑜 mVar 𝑅)‘∅))
16	7, 15	pm2.61i 176	1 ⊢ 𝑋 = ((1𝑜 mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349  ∅c0 4061  ifcif 4223   ↦ cmpt 4861  ^◡ccnv 5248   “ cima 5252  ‘cfv 6031  (class class class)co 6792  1_𝑜c1o 7705   ↑_𝑚 cmap 8008  Fincfn 8108  0cc0 10137  1c1 10138  ℕcn 11221  ℕ₀cn0 11493  0_gc0g 16307  1_rcur 18708   mVar cmvr 19566  var₁cv1 19760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-mvr 19571  df-vr1 19765

This theorem is referenced by:  vr1cl2  19777  vr1cl  19801  subrgvr1  19845  subrgvr1cl  19846  coe1tm  19857  ply1coe  19880  evl1var  19914  evls1var  19916