Theorem mvrvalind 33729

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 21964	. 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 4725	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝐼)
13		indval 12160	. . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ {𝑋} ⊆ 𝐼) → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
14	5, 12, 13	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((𝟭‘𝐼)‘{𝑋}) = (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)))
15		velsn 4578	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋)
16	15	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑋} ↔ 𝑦 = 𝑋))
17	16	ifbid 4485	. . . . . 6 ⊢ (𝜑 → if(𝑦 ∈ {𝑋}, 1, 0) = if(𝑦 = 𝑋, 1, 0))
18	17	mpteq2dv 5173	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐼 ↦ if(𝑦 ∈ {𝑋}, 1, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
19	11, 14, 18	3eqtrd 2779	. . . 4 ⊢ (𝜑 → 𝐴 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))
20	19	eqeq2d 2751	. . 3 ⊢ (𝜑 → (𝐹 = 𝐴 ↔ 𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))))
21	20	ifbid 4485	. 2 ⊢ (𝜑 → if(𝐹 = 𝐴, 1 , 0 ) = if(𝐹 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), 1 , 0 ))
22	9, 21	eqtr4d 2778	1 ⊢ (𝜑 → ((𝑉‘𝑋)‘𝐹) = if(𝐹 = 𝐴, 1 , 0 ))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {crab 3392   ⊆ wss 3890  ifcif 4461  {csn 4562   ↦ cmpt 5160  ^◡ccnv 5624   “ cima 5628  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  Fincfn 8890  0cc0 11036  1c1 11037  𝟭cind 12157  ℕcn 12172  ℕ₀cn0 12435  0_gc0g 17400  1_rcur 20160   mVar cmvr 21887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-oprab 7367  df-mpo 7368  df-ind 12158  df-mvr 21892

This theorem is referenced by:  mplmulmvr  33730