Theorem mvrvalind 33845

Description: Value of the generating elements of the power series structure, expressed using the indicator function. (Contributed by Thierry Arnoux, 25-Jan-2026.)

Hypotheses

Ref	Expression
mvrvalind.1	⊢ 𝑉 = (𝐼 mVar 𝑅)
mvrvalind.2	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
mvrvalind.3	⊢ 0 = (0g‘𝑅)
mvrvalind.4	⊢ 1 = (1r‘𝑅)
mvrvalind.5	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mvrvalind.6	⊢ (𝜑 → 𝑅 ∈ 𝑌)
mvrvalind.7	⊢ (𝜑 → 𝑋 ∈ 𝐼)
mvrvalind.8	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mvrvalind.9	⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋})

Assertion

Ref	Expression
mvrvalind	⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

Distinct variable groups:   ℎ,𝐼   ℎ,𝑋

Allowed substitution hints:   𝜑(ℎ)   𝐴(ℎ)   𝐷(ℎ)   𝑅(ℎ)   1 (ℎ)   𝐹(ℎ)   𝑉(ℎ)   𝑊(ℎ)   𝑌(ℎ)   0 (ℎ)

Proof of Theorem mvrvalind

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mvrvalind.1	. . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅)
2		mvrvalind.2	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
3		mvrvalind.3	. . 3 ⊢ 0 = (0g‘𝑅)
4		mvrvalind.4	. . 3 ⊢ 1 = (1r‘𝑅)
5		mvrvalind.5	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
6		mvrvalind.6	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑌)
7		mvrvalind.7	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
8		mvrvalind.8	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
9	1, 2, 3, 4, 5, 6, 7, 8	mvrval2 22092	. 2 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
10		mvrvalind.9	. . . . . 6 ⊢ 𝐴 = ((𝟭‘𝐼)‘{𝑋})
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐴 = ((𝟭‘𝐼)‘{𝑋}))
12	7	snssd 4748	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐼)
13		indval 12212	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
14	5, 12, 13	syl2anc 595	. . . . 5 ⊢ (𝜑 → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
15		velsn 4601	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋))
17	16	ifbid 4507	. . . . . 6 ⊢ (𝜑 → if(𝑦 ∈ {𝑋}, 1, 0) = if(𝑦 = 𝑋, 1, 0))
18	17	mpteq2dv 5199	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
19	11, 14, 18	3eqtrd 2804	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
20	19	eqeq2d 2776	. . 3 ⊢ (𝜑 → (𝐹 = 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
21	20	ifbid 4507	. 2 ⊢ (𝜑 → if(𝐹 = 𝐴, 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
22	9, 21	eqtr4d 2803	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145  {crab 3417   ⊆ wss 3907  ifcif 4483  {csn 4585   ↦ cmpt 5186  ^◡ccnv 5651   “ cima 5655  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812  Fincfn 8931  0cc0 11088  1c1 11089  𝟭cind 12209  ℕcn 12224  ℕ₀cn0 12495  0_gc0g 17482  1_rcur 20254   mVar cmvr 22015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-ind 12210  df-mvr 22020

This theorem is referenced by:  mplmulmvr  33846