Theorem vr1val 22320

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 7419	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
3	2	fveq1d 6884	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4		df-vr1 22309	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5		fvex 6895	. . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6990	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅))
7	1, 6	eqtrid 2816	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8		fvprc 6874	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6923	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2829	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 22028	. . . . . 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 7545	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 7451	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
14	13	fveq1d 6884	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2807	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
16	7, 15	pm2.61i 184	1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463  ∅c0 4294  ifcif 4492   ↦ cmpt 5196  ^◡ccnv 5661   “ cima 5665  ‘cfv 6537  (class class class)co 7411  1_oc1o 8445   ↑_m cmap 8823  Fincfn 8942  0cc0 11099  1c1 11100  ℕcn 12232  ℕ₀cn0 12503  0_gc0g 17491  1_rcur 20262   mVar cmvr 22023  var₁cv1 22304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-mvr 22028  df-vr1 22309

This theorem is referenced by:  vr1cl2  22321  vr1cl  22345  subrgvr1  22390  subrgvr1cl  22391  coe1tm  22402  ply1coe  22426  evl1var  22464  evls1var  22466  rhmply1vr1  22512