Theorem vr1val 22352

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 7418	. . . . 5 ⊢ (𝑟 = 𝑅 → (1o mVar 𝑟) = (1o mVar 𝑅))
3	2	fveq1d 6883	. . . 4 ⊢ (𝑟 = 𝑅 → ((1o mVar 𝑟)‘∅) = ((1o mVar 𝑅)‘∅))
4		df-vr1 22341	. . . 4 ⊢ var1 = (𝑟 ∈ V ↦ ((1o mVar 𝑟)‘∅))
5		fvex 6894	. . . 4 ⊢ ((1o mVar 𝑅)‘∅) ∈ V
6	3, 4, 5	fvmpt 6989	. . 3 ⊢ (𝑅 ∈ V → (var1‘𝑅) = ((1o mVar 𝑅)‘∅))
7	1, 6	eqtrid 2810	. 2 ⊢ (𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
8		fvprc 6873	. . . 4 ⊢ (¬ 𝑅 ∈ V → (var1‘𝑅) = ∅)
9		0fv 6922	. . . 4 ⊢ (∅‘∅) = ∅
10	8, 1, 9	3eqtr4g 2823	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑋 = (∅‘∅))
11		df-mvr 22060	. . . . . 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 7544	. . . . 5 ⊢ Rel dom mVar
13	12	ovprc2 7450	. . . 4 ⊢ (¬ 𝑅 ∈ V → (1o mVar 𝑅) = ∅)
14	13	fveq1d 6883	. . 3 ⊢ (¬ 𝑅 ∈ V → ((1o mVar 𝑅)‘∅) = (∅‘∅))
15	10, 14	eqtr4d 2801	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑋 = ((1o mVar 𝑅)‘∅))
16	7, 15	pm2.61i 184	1 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ∅c0 4286  ifcif 4487   ↦ cmpt 5192  ^◡ccnv 5660   “ cima 5664  ‘cfv 6536  (class class class)co 7410  1_oc1o 8442   ↑_m cmap 8820  Fincfn 8939  0cc0 11095  1c1 11096  ℕcn 12228  ℕ₀cn0 12499  0_gc0g 17487  1_rcur 20258   mVar cmvr 22055  var₁cv1 22336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-mvr 22060  df-vr1 22341

This theorem is referenced by:  vr1cl2  22353  vr1cl  22377  subrgvr1  22422  subrgvr1cl  22423  coe1tm  22434  ply1coe  22458  evl1var  22496  evls1var  22498  rhmply1vr1  22544