Theorem mhpval 22134

Description: Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.)

Hypotheses

Ref	Expression
mhpfval.h	⊢ 𝐻 = (𝐼 mHomP 𝑅)
mhpfval.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mhpfval.b	⊢ 𝐵 = (Base‘𝑃)
mhpfval.0	⊢ 0 = (0g‘𝑅)
mhpfval.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
mhpfval.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhpfval.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)
mhpval.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)

Assertion

Ref	Expression
mhpval	⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})

Distinct variable groups:   𝑓,𝑔,ℎ   𝑓,𝐼,ℎ   𝑅,𝑓   𝐷,𝑔   𝐵,𝑓   𝑓,𝑁,𝑔

Allowed substitution hints:   𝜑(𝑓,𝑔,ℎ)   𝐵(𝑔,ℎ)   𝐷(𝑓,ℎ)   𝑃(𝑓,𝑔,ℎ)   𝑅(𝑔,ℎ)   𝐻(𝑓,𝑔,ℎ)   𝐼(𝑔)   𝑁(ℎ)   𝑉(𝑓,𝑔,ℎ)   𝑊(𝑓,𝑔,ℎ)   0 (𝑓,𝑔,ℎ)

Proof of Theorem mhpval

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhpfval.h	. . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅)
2		mhpfval.p	. . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
3		mhpfval.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
4		mhpfval.0	. . 3 ⊢ 0 = (0g‘𝑅)
5		mhpfval.d	. . 3 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
6		mhpfval.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
7		mhpfval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
8	1, 2, 3, 4, 5, 6, 7	mhpfval 22133	. 2 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
9		eqeq2 2738	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛 ↔ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁))
10	9	rabbidv 3427	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})
11	10	sseq2d 4012	. . . 4 ⊢ (𝑛 = 𝑁 → ((𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}))
12	11	rabbidv 3427	. . 3 ⊢ (𝑛 = 𝑁 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})
13	12	adantl 480	. 2 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})
14		mhpval.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
15	3	fvexi 6915	. . . 4 ⊢ 𝐵 ∈ V
16	15	rabex 5339	. . 3 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}} ∈ V
17	16	a1i 11	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}} ∈ V)
18	8, 13, 14, 17	fvmptd 7016	1 ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  {crab 3419  Vcvv 3462   ⊆ wss 3947  ^◡ccnv 5681   “ cima 5685  ‘cfv 6554  (class class class)co 7424   supp csupp 8174   ↑_m cmap 8855  Fincfn 8974  ℕcn 12264  ℕ₀cn0 12524  Basecbs 17213   ↾_s cress 17242  0_gc0g 17454   Σ_g cgsu 17455  ℂ_fldccnfld 21343   mPoly cmpl 21903   mHomP cmhp 22124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-1cn 11216  ax-addcl 11218

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-nn 12265  df-n0 12525  df-mhp 22131

This theorem is referenced by:  ismhp  22135