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Mirrors > Home > MPE Home > Th. List > mhpval | Structured version Visualization version GIF version |
Description: Value of the "homogeneous polynomial" function. (Contributed by Steven Nguyen, 25-Aug-2023.) |
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) |
Ref | Expression |
---|---|
mhpval | ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}}) |
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 20332 | . 2 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ ℕ0 ↦ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}})) |
9 | eqeq2 2833 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛 ↔ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁)) | |
10 | 9 | rabbidv 3480 | . . . . 5 ⊢ (𝑛 = 𝑁 → {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛} = {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}) |
11 | 10 | sseq2d 3999 | . . . 4 ⊢ (𝑛 = 𝑁 → ((𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛} ↔ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁})) |
12 | 11 | rabbidv 3480 | . . 3 ⊢ (𝑛 = 𝑁 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}}) |
13 | 12 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑛}} = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}}) |
14 | mhpval.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
15 | 3 | fvexi 6684 | . . . 4 ⊢ 𝐵 ∈ V |
16 | 15 | rabex 5235 | . . 3 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}} ∈ V |
17 | 16 | a1i 11 | . 2 ⊢ (𝜑 → {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}} ∈ V) |
18 | 8, 13, 14, 17 | fvmptd 6775 | 1 ⊢ (𝜑 → (𝐻‘𝑁) = {𝑓 ∈ 𝐵 ∣ (𝑓 supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ⊆ wss 3936 ◡ccnv 5554 “ cima 5558 ‘cfv 6355 (class class class)co 7156 supp csupp 7830 ↑m cmap 8406 Fincfn 8509 ℕcn 11638 ℕ0cn0 11898 Σcsu 15042 Basecbs 16483 0gc0g 16713 mPoly cmpl 20133 mHomP cmhp 20322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-1cn 10595 ax-addcl 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-n0 11899 df-mhp 20326 |
This theorem is referenced by: ismhp 20334 |
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