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Mirrors > Home > MPE Home > Th. List > evlslem6 | Structured version Visualization version GIF version |
Description: Lemma for evlseu 19876. 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 18912 | . . . . . 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 19082 | . . . . . . 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 19794 | . . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶𝐾) |
16 | 15 | ffvelrnda 6608 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ 𝐾) |
17 | 10, 16 | ffvelrnd 6609 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶) |
18 | evlslem1.t | . . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆) | |
19 | 18, 7 | mgpbas 18849 | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) |
20 | evlslem1.x | . . . . 5 ⊢ ↑ = (.g‘𝑇) | |
21 | eqid 2825 | . . . . 5 ⊢ (0g‘𝑇) = (0g‘𝑇) | |
22 | 18 | crngmgp 18909 | . . . . . . 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 19871 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)) ∈ 𝐶) |
31 | evlslem1.m | . . . . 5 ⊢ · = (.r‘𝑆) | |
32 | 7, 31 | ringcl 18915 | . . . 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 1496 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺))) ∈ 𝐶) |
34 | 33 | fmpttd 6634 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))):𝐷⟶𝐶) |
35 | ovexd 6939 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑𝑚 𝐼) ∈ V) | |
36 | 13, 35 | rabexd 5038 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
37 | mptexg 6740 | . . . 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 6161 | . . . 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 6448 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
42 | eqid 2825 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
43 | evlslem1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
44 | 11, 12, 42, 14, 43 | mplelsfi 19851 | . . . 4 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅)) |
45 | 44 | fsuppimpd 8551 | . . 3 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin) |
46 | 15 | feqmptd 6496 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏))) |
47 | 46 | oveq1d 6920 | . . . . . 6 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅))) |
48 | eqimss2 3883 | . . . . . 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 19081 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
51 | eqid 2825 | . . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
52 | 42, 51 | ghmid 18017 | . . . . . 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 6448 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V) | |
55 | fvexd 6448 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
56 | 49, 53, 54, 55 | suppssfv 7596 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
57 | 7, 31, 51 | ringlz 18941 | . . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
58 | 3, 57 | sylan 577 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
59 | fvexd 6448 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V) | |
60 | 56, 58, 59, 30, 41 | suppssov1 7592 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
61 | suppssfifsupp 8559 | . . 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 1503 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘𝑓 ↑ 𝐺)))) finSupp (0g‘𝑆)) |
63 | 34, 62 | jca 509 | 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 1658 ∈ wcel 2166 {crab 3121 Vcvv 3414 ⊆ wss 3798 class class class wbr 4873 ↦ cmpt 4952 ◡ccnv 5341 “ cima 5345 Fun wfun 6117 ⟶wf 6119 ‘cfv 6123 (class class class)co 6905 ∘𝑓 cof 7155 supp csupp 7559 ↑𝑚 cmap 8122 Fincfn 8222 finSupp cfsupp 8544 ℕcn 11350 ℕ0cn0 11618 Basecbs 16222 .rcmulr 16306 0gc0g 16453 Σg cgsu 16454 .gcmg 17894 GrpHom cghm 18008 CMndccmn 18546 mulGrpcmgp 18843 Ringcrg 18901 CRingccrg 18902 RingHom crh 19068 mVar cmvr 19713 mPoly cmpl 19714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-oi 8684 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-seq 13096 df-hash 13411 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-tset 16324 df-0g 16455 df-gsum 16456 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-mhm 17688 df-grp 17779 df-minusg 17780 df-mulg 17895 df-ghm 18009 df-cntz 18100 df-cmn 18548 df-mgp 18844 df-ur 18856 df-ring 18903 df-cring 18904 df-rnghom 19071 df-psr 19717 df-mpl 19719 |
This theorem is referenced by: evlslem1 19875 |
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