Theorem mhpval 22191

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 22190	. 2 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
9		eqeq2 2773	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛 ↔ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁))
10	9	rabbidv 3420	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})
11	10	sseq2d 3966	. . . 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 3420	. . 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 485	. 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 6875	. . . 4 ⊢ 𝐵 ∈ V
16	15	rabex 5292	. . 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 6977	1 ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ⊆ wss 3902  ^◡ccnv 5642   “ cima 5646  ‘cfv 6515  (class class class)co 7390   supp csupp 8133   ↑_m cmap 8801  Fincfn 8920  ℕcn 12203  ℕ₀cn0 12474  Basecbs 17235   ↾_s cress 17256  0_gc0g 17458   Σ_g cgsu 17459  ℂ_fldccnfld 21411   mPoly cmpl 21945   mHomP cmhp 22185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-1cn 11124  ax-addcl 11126

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-nn 12204  df-n0 12475  df-mhp 22188

This theorem is referenced by:  ismhp  22192