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Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvvvallem2 | Structured version Visualization version GIF version |
Description: Lemma for theorems using evlsvvval 41590. (Contributed by SN, 8-Mar-2025.) |
Ref | Expression |
---|---|
evlsvvvallem2.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
evlsvvvallem2.p | ⊢ 𝑃 = (𝐼 mPoly 𝑈) |
evlsvvvallem2.u | ⊢ 𝑈 = (𝑆 ↾s 𝑅) |
evlsvvvallem2.b | ⊢ 𝐵 = (Base‘𝑃) |
evlsvvvallem2.k | ⊢ 𝐾 = (Base‘𝑆) |
evlsvvvallem2.m | ⊢ 𝑀 = (mulGrp‘𝑆) |
evlsvvvallem2.w | ⊢ ↑ = (.g‘𝑀) |
evlsvvvallem2.x | ⊢ · = (.r‘𝑆) |
evlsvvvallem2.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
evlsvvvallem2.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlsvvvallem2.r | ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) |
evlsvvvallem2.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
evlsvvvallem2.a | ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
Ref | Expression |
---|---|
evlsvvvallem2 | ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlsvvvallem2.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
2 | ovex 7434 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
3 | 1, 2 | rabex2 5324 | . . . 4 ⊢ 𝐷 ∈ V |
4 | 3 | mptex 7216 | . . 3 ⊢ (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) ∈ V |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) ∈ V) |
6 | fvexd 6896 | . 2 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
7 | funmpt 6576 | . . 3 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))))) |
9 | evlsvvvallem2.p | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑈) | |
10 | evlsvvvallem2.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
11 | eqid 2724 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
12 | evlsvvvallem2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
13 | evlsvvvallem2.u | . . . . 5 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
14 | 13 | ovexi 7435 | . . . 4 ⊢ 𝑈 ∈ V |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
16 | 9, 10, 11, 12, 15 | mplelsfi 21855 | . 2 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑈)) |
17 | eqid 2724 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
18 | 9, 17, 10, 1, 12 | mplelf 21858 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈)) |
19 | ssidd 3997 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ (𝐹 supp (0g‘𝑈))) | |
20 | fvexd 6896 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑈) ∈ V) | |
21 | 18, 19, 12, 20 | suppssrg 8176 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝐹‘𝑏) = (0g‘𝑈)) |
22 | evlsvvvallem2.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
23 | eqid 2724 | . . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
24 | 13, 23 | subrg0 20466 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈)) |
25 | 22, 24 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈)) |
26 | 25 | eqcomd 2730 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑈) = (0g‘𝑆)) |
27 | 26 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (0g‘𝑈) = (0g‘𝑆)) |
28 | 21, 27 | eqtrd 2764 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝐹‘𝑏) = (0g‘𝑆)) |
29 | 28 | oveq1d 7416 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = ((0g‘𝑆) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) |
30 | evlsvvvallem2.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
31 | evlsvvvallem2.x | . . . . 5 ⊢ · = (.r‘𝑆) | |
32 | evlsvvvallem2.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
33 | 32 | crngringd 20136 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring) |
34 | 33 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → 𝑆 ∈ Ring) |
35 | eldifi 4118 | . . . . . 6 ⊢ (𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈))) → 𝑏 ∈ 𝐷) | |
36 | evlsvvvallem2.m | . . . . . . 7 ⊢ 𝑀 = (mulGrp‘𝑆) | |
37 | evlsvvvallem2.w | . . . . . . 7 ⊢ ↑ = (.g‘𝑀) | |
38 | evlsvvvallem2.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
39 | 38 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉) |
40 | 32 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing) |
41 | evlsvvvallem2.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
42 | 41 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑m 𝐼)) |
43 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
44 | 1, 30, 36, 37, 39, 40, 42, 43 | evlsvvvallem 41588 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) |
45 | 35, 44 | sylan2 592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) |
46 | 30, 31, 23, 34, 45 | ringlzd 20179 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → ((0g‘𝑆) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0g‘𝑆)) |
47 | 29, 46 | eqtrd 2764 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0g‘𝑆)) |
48 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) |
49 | 47, 48 | suppss2 8180 | . 2 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) supp (0g‘𝑆)) ⊆ (𝐹 supp (0g‘𝑈))) |
50 | 5, 6, 8, 16, 49 | fsuppsssuppgd 41523 | 1 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) finSupp (0g‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ∖ cdif 3937 class class class wbr 5138 ↦ cmpt 5221 ◡ccnv 5665 “ cima 5669 Fun wfun 6527 ‘cfv 6533 (class class class)co 7401 supp csupp 8140 ↑m cmap 8815 Fincfn 8934 finSupp cfsupp 9356 ℕcn 12208 ℕ0cn0 12468 Basecbs 17140 ↾s cress 17169 .rcmulr 17194 0gc0g 17381 Σg cgsu 17382 .gcmg 18982 mulGrpcmgp 20024 Ringcrg 20123 CRingccrg 20124 SubRingcsubrg 20454 mPoly cmpl 21759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-oi 9500 df-card 9929 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-tset 17212 df-0g 17383 df-gsum 17384 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-grp 18853 df-minusg 18854 df-mulg 18983 df-subg 19035 df-cntz 19218 df-cmn 19687 df-abl 19688 df-mgp 20025 df-rng 20043 df-ur 20072 df-ring 20125 df-cring 20126 df-subrg 20456 df-psr 21762 df-mpl 21764 |
This theorem is referenced by: evlsbagval 41593 evlvvvallem 41601 evlsmhpvvval 41622 mhphf 41624 |
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