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| Mirrors > Home > MPE Home > Th. List > Mathboxes > evlsvvvallem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for theorems using evlsvvval 42573. (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 7464 | . . . . 5 ⊢ (ℕ0 ↑m 𝐼) ∈ V | |
| 3 | 1, 2 | rabex2 5341 | . . . 4 ⊢ 𝐷 ∈ V | 
| 4 | 3 | mptex 7243 | . . 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 6921 | . 2 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
| 7 | funmpt 6604 | . . 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 2737 | . . 3 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
| 12 | evlsvvvallem2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
| 13 | 9, 10, 11, 12 | mplelsfi 22015 | . 2 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑈)) | 
| 14 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 15 | 9, 14, 10, 1, 12 | mplelf 22018 | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐷⟶(Base‘𝑈)) | 
| 16 | ssidd 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐹 supp (0g‘𝑈)) ⊆ (𝐹 supp (0g‘𝑈))) | |
| 17 | fvexd 6921 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑈) ∈ V) | |
| 18 | 15, 16, 12, 17 | suppssrg 8221 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝐹‘𝑏) = (0g‘𝑈)) | 
| 19 | evlsvvvallem2.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆)) | |
| 20 | evlsvvvallem2.u | . . . . . . . . . 10 ⊢ 𝑈 = (𝑆 ↾s 𝑅) | |
| 21 | eqid 2737 | . . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
| 22 | 20, 21 | subrg0 20579 | . . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (0g‘𝑆) = (0g‘𝑈)) | 
| 23 | 19, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑈)) | 
| 24 | 23 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → (0g‘𝑈) = (0g‘𝑆)) | 
| 25 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (0g‘𝑈) = (0g‘𝑆)) | 
| 26 | 18, 25 | eqtrd 2777 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝐹‘𝑏) = (0g‘𝑆)) | 
| 27 | 26 | oveq1d 7446 | . . . 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)))))) | 
| 28 | evlsvvvallem2.k | . . . . 5 ⊢ 𝐾 = (Base‘𝑆) | |
| 29 | evlsvvvallem2.x | . . . . 5 ⊢ · = (.r‘𝑆) | |
| 30 | evlsvvvallem2.s | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
| 31 | 30 | crngringd 20243 | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Ring) | 
| 32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → 𝑆 ∈ Ring) | 
| 33 | eldifi 4131 | . . . . . 6 ⊢ (𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈))) → 𝑏 ∈ 𝐷) | |
| 34 | evlsvvvallem2.m | . . . . . . 7 ⊢ 𝑀 = (mulGrp‘𝑆) | |
| 35 | evlsvvvallem2.w | . . . . . . 7 ⊢ ↑ = (.g‘𝑀) | |
| 36 | evlsvvvallem2.i | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
| 37 | 36 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑉) | 
| 38 | 30 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ CRing) | 
| 39 | evlsvvvallem2.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑m 𝐼)) | |
| 40 | 39 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐴 ∈ (𝐾 ↑m 𝐼)) | 
| 41 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
| 42 | 1, 28, 34, 35, 37, 38, 40, 41 | evlsvvvallem 42571 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) | 
| 43 | 33, 42 | sylan2 593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))) ∈ 𝐾) | 
| 44 | 28, 29, 21, 32, 43 | ringlzd 20292 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → ((0g‘𝑆) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0g‘𝑆)) | 
| 45 | 27, 44 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (𝐷 ∖ (𝐹 supp (0g‘𝑈)))) → ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣))))) = (0g‘𝑆)) | 
| 46 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐷 ∈ V) | 
| 47 | 45, 46 | suppss2 8225 | . 2 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘𝑏) · (𝑀 Σg (𝑣 ∈ 𝐼 ↦ ((𝑏‘𝑣) ↑ (𝐴‘𝑣)))))) supp (0g‘𝑆)) ⊆ (𝐹 supp (0g‘𝑈))) | 
| 48 | 5, 6, 8, 13, 47 | fsuppsssuppgd 9422 | 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 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∖ cdif 3948 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 0gc0g 17484 Σg cgsu 17485 .gcmg 19085 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 SubRingcsubrg 20569 mPoly cmpl 21926 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-tset 17316 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrg 20570 df-psr 21929 df-mpl 21931 | 
| This theorem is referenced by: evlsbagval 42576 evlvvvallem 42584 evlsmhpvvval 42605 mhphf 42607 | 
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