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Mirrors > Home > MPE Home > Th. List > evlslem6 | Structured version Visualization version GIF version |
Description: Lemma for evlseu 19912. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.) |
Ref | Expression |
---|---|
evlslem1.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
evlslem1.b | ⊢ 𝐵 = (Base‘𝑃) |
evlslem1.c | ⊢ 𝐶 = (Base‘𝑆) |
evlslem1.k | ⊢ 𝐾 = (Base‘𝑅) |
evlslem1.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
evlslem1.t | ⊢ 𝑇 = (mulGrp‘𝑆) |
evlslem1.x | ⊢ ↑ = (.g‘𝑇) |
evlslem1.m | ⊢ · = (.r‘𝑆) |
evlslem1.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
evlslem1.e | ⊢ 𝐸 = (𝑝 ∈ 𝐵 ↦ (𝑆 Σg (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑝‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))))) |
evlslem1.i | ⊢ (𝜑 → 𝐼 ∈ V) |
evlslem1.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
evlslem1.s | ⊢ (𝜑 → 𝑆 ∈ CRing) |
evlslem1.f | ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) |
evlslem1.g | ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) |
evlslem6.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
evlslem6 | ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) finSupp (0g‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evlslem1.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing) | |
2 | crngring 18945 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
4 | 3 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) |
5 | evlslem1.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
6 | evlslem1.k | . . . . . . . 8 ⊢ 𝐾 = (Base‘𝑅) | |
7 | evlslem1.c | . . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆) | |
8 | 6, 7 | rhmf 19115 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐾⟶𝐶) |
9 | 5, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐾⟶𝐶) |
10 | 9 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐹:𝐾⟶𝐶) |
11 | evlslem1.p | . . . . . . 7 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
12 | evlslem1.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑃) | |
13 | evlslem1.d | . . . . . . 7 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
14 | evlslem6.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
15 | 11, 6, 12, 13, 14 | mplelf 19830 | . . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶𝐾) |
16 | 15 | ffvelrnda 6623 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ 𝐾) |
17 | 10, 16 | ffvelrnd 6624 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶) |
18 | evlslem1.t | . . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆) | |
19 | 18, 7 | mgpbas 18882 | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) |
20 | evlslem1.x | . . . . 5 ⊢ ↑ = (.g‘𝑇) | |
21 | eqid 2778 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
22 | 18 | crngmgp 18942 | . . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd) |
23 | 1, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd) |
24 | 23 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd) |
25 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
26 | evlslem1.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
27 | 26 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶) |
28 | evlslem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) | |
29 | 28 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ V) |
30 | 13, 19, 20, 21, 24, 25, 27, 29 | psrbagev2 19907 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)) ∈ 𝐶) |
31 | evlslem1.m | . . . . 5 ⊢ · = (.r‘𝑆) | |
32 | 7, 31 | ringcl 18948 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘(𝑌‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)) ∈ 𝐶) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺))) ∈ 𝐶) |
33 | 4, 17, 30, 32 | syl3anc 1439 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺))) ∈ 𝐶) |
34 | 33 | fmpttd 6649 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))):𝐷⟶𝐶) |
35 | ovexd 6956 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V) | |
36 | 13, 35 | rabexd 5050 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
37 | mptexg 6756 | . . . 4 ⊢ (𝐷 ∈ V → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) ∈ V) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) ∈ V) |
39 | funmpt 6173 | . . . 4 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺))))) |
41 | fvexd 6461 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
42 | eqid 2778 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
43 | evlslem1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
44 | 11, 12, 42, 14, 43 | mplelsfi 19887 | . . . 4 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅)) |
45 | 44 | fsuppimpd 8570 | . . 3 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin) |
46 | 15 | feqmptd 6509 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏))) |
47 | 46 | oveq1d 6937 | . . . . . 6 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅))) |
48 | eqimss2 3877 | . . . . . 6 ⊢ ((𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) | |
49 | 47, 48 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) |
50 | rhmghm 19114 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
51 | eqid 2778 | . . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
52 | 42, 51 | ghmid 18050 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
53 | 5, 50, 52 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
54 | fvexd 6461 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V) | |
55 | fvexd 6461 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
56 | 49, 53, 54, 55 | suppssfv 7613 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
57 | 7, 31, 51 | ringlz 18974 | . . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
58 | 3, 57 | sylan 575 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
59 | fvexd 6461 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V) | |
60 | 56, 58, 59, 30, 41 | suppssov1 7609 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
61 | suppssfifsupp 8578 | . . 3 ⊢ ((((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) ∈ V ∧ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) ∧ (0g‘𝑆) ∈ V) ∧ ((𝑌 supp (0g‘𝑅)) ∈ Fin ∧ ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅)))) → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) finSupp (0g‘𝑆)) | |
62 | 38, 40, 41, 45, 60, 61 | syl32anc 1446 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) finSupp (0g‘𝑆)) |
63 | 34, 62 | jca 507 | 1 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))):𝐷⟶𝐶 ∧ (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) finSupp (0g‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 {crab 3094 Vcvv 3398 ⊆ wss 3792 class class class wbr 4886 ↦ cmpt 4965 ◡ccnv 5354 “ cima 5358 Fun wfun 6129 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 supp csupp 7576 ↑𝑚 cmap 8140 Fincfn 8241 finSupp cfsupp 8563 ℕcn 11374 ℕ0cn0 11642 Basecbs 16255 .rcmulr 16339 0gc0g 16486 Σg cgsu 16487 .gcmg 17927 GrpHom cghm 18041 CMndccmn 18579 mulGrpcmgp 18876 Ringcrg 18934 CRingccrg 18935 RingHom crh 19101 mVar cmvr 19749 mPoly cmpl 19750 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-tset 16357 df-0g 16488 df-gsum 16489 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-mulg 17928 df-ghm 18042 df-cntz 18133 df-cmn 18581 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-rnghom 19104 df-psr 19753 df-mpl 19755 |
This theorem is referenced by: evlslem1 19911 |
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