Theorem mvrvalind 33959

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 22169	. 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 4757	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐼)
13		indval 12239	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
14	5, 12, 13	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
15		velsn 4610	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋))
17	16	ifbid 4516	. . . . . 6 ⊢ (𝜑 → if(𝑦 ∈ {𝑋}, 1, 0) = if(𝑦 = 𝑋, 1, 0))
18	17	mpteq2dv 5210	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
19	11, 14, 18	3eqtrd 2805	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
20	19	eqeq2d 2777	. . 3 ⊢ (𝜑 → (𝐹 = 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
21	20	ifbid 4516	. 2 ⊢ (𝜑 → if(𝐹 = 𝐴, 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
22	9, 21	eqtr4d 2804	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  {crab 3419   ⊆ wss 3908  ifcif 4492  {csn 4594   ↦ cmpt 5197  ^◡ccnv 5665   “ cima 5669  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833  Fincfn 8952  0cc0 11118  1c1 11119  𝟭cind 12236  ℕcn 12251  ℕ₀cn0 12522  0_gc0g 17517  1_rcur 20294   mVar cmvr 22092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-ind 12237  df-mvr 22097

This theorem is used by:  mplmulmvr  33960