Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mhp0cl | Structured version Visualization version GIF version |
Description: The zero polynomial is homogeneous. (Contributed by SN, 12-Sep-2023.) |
Ref | Expression |
---|---|
mhp0cl.h | ⊢ 𝐻 = (𝐼 mHomP 𝑅) |
mhp0cl.0 | ⊢ 0 = (0g‘𝑅) |
mhp0cl.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
mhp0cl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
mhp0cl.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
mhp0cl.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
mhp0cl | ⊢ (𝜑 → (𝐷 × { 0 }) ∈ (𝐻‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2820 | . . . 4 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅) | |
2 | mhp0cl.d | . . . 4 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | mhp0cl.0 | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | eqid 2820 | . . . 4 ⊢ (0g‘(𝐼 mPoly 𝑅)) = (0g‘(𝐼 mPoly 𝑅)) | |
5 | mhp0cl.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
6 | mhp0cl.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) | |
7 | 1, 2, 3, 4, 5, 6 | mpl0 20214 | . . 3 ⊢ (𝜑 → (0g‘(𝐼 mPoly 𝑅)) = (𝐷 × { 0 })) |
8 | 1 | mplgrp 20223 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp) → (𝐼 mPoly 𝑅) ∈ Grp) |
9 | 5, 6, 8 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐼 mPoly 𝑅) ∈ Grp) |
10 | eqid 2820 | . . . . 5 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅)) | |
11 | 10, 4 | grpidcl 18124 | . . . 4 ⊢ ((𝐼 mPoly 𝑅) ∈ Grp → (0g‘(𝐼 mPoly 𝑅)) ∈ (Base‘(𝐼 mPoly 𝑅))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → (0g‘(𝐼 mPoly 𝑅)) ∈ (Base‘(𝐼 mPoly 𝑅))) |
13 | 7, 12 | eqeltrrd 2913 | . 2 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ (Base‘(𝐼 mPoly 𝑅))) |
14 | fczsupp0 7852 | . . . 4 ⊢ ((𝐷 × { 0 }) supp 0 ) = ∅ | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) = ∅) |
16 | 0ss 4343 | . . 3 ⊢ ∅ ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁} | |
17 | 15, 16 | eqsstrdi 4014 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}) |
18 | mhp0cl.h | . . 3 ⊢ 𝐻 = (𝐼 mHomP 𝑅) | |
19 | mhp0cl.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
20 | 18, 1, 10, 3, 2, 5, 6, 19 | ismhp 20327 | . 2 ⊢ (𝜑 → ((𝐷 × { 0 }) ∈ (𝐻‘𝑁) ↔ ((𝐷 × { 0 }) ∈ (Base‘(𝐼 mPoly 𝑅)) ∧ ((𝐷 × { 0 }) supp 0 ) ⊆ {𝑔 ∈ 𝐷 ∣ Σ𝑗 ∈ ℕ0 (𝑔‘𝑗) = 𝑁}))) |
21 | 13, 17, 20 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝐷 × { 0 }) ∈ (𝐻‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3141 ⊆ wss 3929 ∅c0 4284 {csn 4560 × cxp 5546 ◡ccnv 5547 “ cima 5551 ‘cfv 6348 (class class class)co 7149 supp csupp 7823 ↑m cmap 8399 Fincfn 8502 ℕcn 11631 ℕ0cn0 11891 Σcsu 15035 Basecbs 16476 0gc0g 16706 Grpcgrp 18096 mPoly cmpl 20126 mHomP cmhp 20315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-of 7402 df-om 7574 df-1st 7682 df-2nd 7683 df-supp 7824 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fsupp 8827 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-mulr 16572 df-sca 16574 df-vsca 16575 df-tset 16577 df-0g 16708 df-mgm 17845 df-sgrp 17894 df-mnd 17905 df-grp 18099 df-minusg 18100 df-subg 18269 df-psr 20129 df-mpl 20131 df-mhp 20319 |
This theorem is referenced by: mhpsubg 20333 |
Copyright terms: Public domain | W3C validator |