Theorem mvrvalind 33703

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 21938	. 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 4765	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐼)
13		indval 32932	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
14	5, 12, 13	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
15		velsn 4596	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋))
17	16	ifbid 4503	. . . . . 6 ⊢ (𝜑 → if(𝑦 ∈ {𝑋}, 1, 0) = if(𝑦 = 𝑋, 1, 0))
18	17	mpteq2dv 5192	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
19	11, 14, 18	3eqtrd 2775	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
20	19	eqeq2d 2747	. . 3 ⊢ (𝜑 → (𝐹 = 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
21	20	ifbid 4503	. 2 ⊢ (𝜑 → if(𝐹 = 𝐴, 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
22	9, 21	eqtr4d 2774	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {crab 3399   ⊆ wss 3901  ifcif 4479  {csn 4580   ↦ cmpt 5179  ^◡ccnv 5623   “ cima 5627  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763  Fincfn 8883  0cc0 11026  1c1 11027  ℕcn 12145  ℕ₀cn0 12401  0_gc0g 17359  1_rcur 20116   mVar cmvr 21861  𝟭cind 32929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mvr 21866  df-ind 32930

This theorem is referenced by:  mplmulmvr  33704