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Mirrors > Home > MPE Home > Th. List > evlslem6 | Structured version Visualization version GIF version |
Description: Lemma for evlseu 21493. Finiteness and consistency of the top-level sum. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 26-Jul-2019.) (Revised by AV, 11-Apr-2024.) |
Ref | Expression |
---|---|
evlslem1.p | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
evlslem1.b | ⊢ 𝐵 = (Base‘𝑃) |
evlslem1.c | ⊢ 𝐶 = (Base‘𝑆) |
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 ∈ 𝑊) |
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 19976 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) |
5 | evlslem1.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
6 | eqid 2736 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | evlslem1.c | . . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆) | |
8 | 6, 7 | rhmf 20158 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐶) |
9 | 5, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶𝐶) |
10 | 9 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐹:(Base‘𝑅)⟶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 21404 | . . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) |
16 | 15 | ffvelcdmda 7035 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅)) |
17 | 10, 16 | ffvelcdmd 7036 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶) |
18 | evlslem1.t | . . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆) | |
19 | 18, 7 | mgpbas 19902 | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) |
20 | evlslem1.x | . . . . 5 ⊢ ↑ = (.g‘𝑇) | |
21 | 18 | crngmgp 19972 | . . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd) |
22 | 1, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd) |
23 | 22 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd) |
24 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
25 | evlslem1.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
26 | 25 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶) |
27 | 13, 19, 20, 23, 24, 26 | psrbagev2 21487 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘f ↑ 𝐺)) ∈ 𝐶) |
28 | evlslem1.m | . . . . 5 ⊢ · = (.r‘𝑆) | |
29 | 7, 28 | ringcl 19981 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘(𝑌‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σg (𝑏 ∘f ↑ 𝐺)) ∈ 𝐶) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))) ∈ 𝐶) |
30 | 4, 17, 27, 29 | syl3anc 1371 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))) ∈ 𝐶) |
31 | 30 | fmpttd 7063 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))):𝐷⟶𝐶) |
32 | ovexd 7392 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
33 | 13, 32 | rabexd 5290 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
34 | 33 | mptexd 7174 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) ∈ V) |
35 | funmpt 6539 | . . . 4 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))))) |
37 | fvexd 6857 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
38 | eqid 2736 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
39 | evlslem1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
40 | 11, 12, 38, 14, 39 | mplelsfi 21401 | . . . 4 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅)) |
41 | 40 | fsuppimpd 9312 | . . 3 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin) |
42 | 15 | feqmptd 6910 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏))) |
43 | 42 | oveq1d 7372 | . . . . . 6 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅))) |
44 | eqimss2 4001 | . . . . . 6 ⊢ ((𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) | |
45 | 43, 44 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) |
46 | rhmghm 20157 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
47 | eqid 2736 | . . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
48 | 38, 47 | ghmid 19014 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
49 | 5, 46, 48 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
50 | fvexd 6857 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V) | |
51 | fvexd 6857 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
52 | 45, 49, 50, 51 | suppssfv 8133 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
53 | 7, 28, 47 | ringlz 20011 | . . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
54 | 3, 53 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
55 | fvexd 6857 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V) | |
56 | 52, 54, 55, 27, 37 | suppssov1 8129 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
57 | suppssfifsupp 9320 | . . 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)) | |
58 | 34, 36, 37, 41, 56, 57 | syl32anc 1378 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) finSupp (0g‘𝑆)) |
59 | 31, 58 | jca 512 | 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 396 = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 ◡ccnv 5632 “ cima 5636 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∘f cof 7615 supp csupp 8092 ↑m cmap 8765 Fincfn 8883 finSupp cfsupp 9305 ℕcn 12153 ℕ0cn0 12413 Basecbs 17083 .rcmulr 17134 0gc0g 17321 Σg cgsu 17322 .gcmg 18872 GrpHom cghm 19005 CMndccmn 19562 mulGrpcmgp 19896 Ringcrg 19964 CRingccrg 19965 RingHom crh 20143 mVar cmvr 21307 mPoly cmpl 21308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-tset 17152 df-0g 17323 df-gsum 17324 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-mulg 18873 df-ghm 19006 df-cntz 19097 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-rnghom 20146 df-psr 21311 df-mpl 21313 |
This theorem is referenced by: evlslem1 21492 |
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