Theorem mhpval 22094

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 22093	. 2 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛}}))
9		eqeq2 2749	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛 ↔ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁))
10	9	rabbidv 3408	. . . . 5 ⊢ (𝑛 = 𝑁 → {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑛} = {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁})
11	10	sseq2d 3968	. . . 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 3408	. . 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 481	. 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 6856	. . . 4 ⊢ 𝐵 ∈ V
16	15	rabex 5286	. . 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 6957	1 ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ ((ℂfld ↾s ℕ0) Σg 𝑔) = 𝑁}})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3401  Vcvv 3442   ⊆ wss 3903  ^◡ccnv 5631   “ cima 5635  ‘cfv 6500  (class class class)co 7368   supp csupp 8112   ↑_m cmap 8775  Fincfn 8895  ℕcn 12157  ℕ₀cn0 12413  Basecbs 17148   ↾_s cress 17169  0_gc0g 17371   Σ_g cgsu 17372  ℂ_fldccnfld 21321   mPoly cmpl 21874   mHomP cmhp 22084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-1cn 11096  ax-addcl 11098

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-nn 12158  df-n0 12414  df-mhp 22091

This theorem is referenced by:  ismhp  22095