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Mirrors > Home > MPE Home > Th. List > evlslem6 | Structured version Visualization version GIF version |
Description: Lemma for evlseu 20755. 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 19302 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) |
5 | evlslem1.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
6 | eqid 2798 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
7 | evlslem1.c | . . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆) | |
8 | 6, 7 | rhmf 19474 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐶) |
9 | 5, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶𝐶) |
10 | 9 | adantr 484 | . . . . 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 20671 | . . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) |
16 | 15 | ffvelrnda 6828 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅)) |
17 | 10, 16 | ffvelrnd 6829 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶) |
18 | evlslem1.t | . . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆) | |
19 | 18, 7 | mgpbas 19238 | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) |
20 | evlslem1.x | . . . . 5 ⊢ ↑ = (.g‘𝑇) | |
21 | 18 | crngmgp 19298 | . . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd) |
22 | 1, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd) |
23 | 22 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd) |
24 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
25 | evlslem1.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
26 | 25 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶) |
27 | evlslem1.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
28 | 27 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐼 ∈ 𝑊) |
29 | 13, 19, 20, 23, 24, 26, 28 | psrbagev2 20750 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘f ↑ 𝐺)) ∈ 𝐶) |
30 | evlslem1.m | . . . . 5 ⊢ · = (.r‘𝑆) | |
31 | 7, 30 | ringcl 19307 | . . . 4 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘(𝑌‘𝑏)) ∈ 𝐶 ∧ (𝑇 Σg (𝑏 ∘f ↑ 𝐺)) ∈ 𝐶) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))) ∈ 𝐶) |
32 | 4, 17, 29, 31 | syl3anc 1368 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))) ∈ 𝐶) |
33 | 32 | fmpttd 6856 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))):𝐷⟶𝐶) |
34 | ovexd 7170 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
35 | 13, 34 | rabexd 5200 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) |
36 | 35 | mptexd 6964 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) ∈ V) |
37 | funmpt 6362 | . . . 4 ⊢ Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝜑 → Fun (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))))) |
39 | fvexd 6660 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
40 | eqid 2798 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
41 | evlslem1.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
42 | 11, 12, 40, 14, 41 | mplelsfi 20730 | . . . 4 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅)) |
43 | 42 | fsuppimpd 8824 | . . 3 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin) |
44 | 15 | feqmptd 6708 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏))) |
45 | 44 | oveq1d 7150 | . . . . . 6 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅))) |
46 | eqimss2 3972 | . . . . . 6 ⊢ ((𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) | |
47 | 45, 46 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) |
48 | rhmghm 19473 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
49 | eqid 2798 | . . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
50 | 40, 49 | ghmid 18356 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
51 | 5, 48, 50 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) |
52 | fvexd 6660 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V) | |
53 | fvexd 6660 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
54 | 47, 51, 52, 53 | suppssfv 7849 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
55 | 7, 30, 49 | ringlz 19333 | . . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
56 | 3, 55 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) |
57 | fvexd 6660 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V) | |
58 | 54, 56, 57, 29, 39 | suppssov1 7845 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) |
59 | suppssfifsupp 8832 | . . 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)) | |
60 | 36, 38, 39, 43, 58, 59 | syl32anc 1375 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) finSupp (0g‘𝑆)) |
61 | 33, 60 | jca 515 | 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 399 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 supp csupp 7813 ↑m cmap 8389 Fincfn 8492 finSupp cfsupp 8817 ℕcn 11625 ℕ0cn0 11885 Basecbs 16475 .rcmulr 16558 0gc0g 16705 Σg cgsu 16706 .gcmg 18216 GrpHom cghm 18347 CMndccmn 18898 mulGrpcmgp 19232 Ringcrg 19290 CRingccrg 19291 RingHom crh 19460 mVar cmvr 20590 mPoly cmpl 20591 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-tset 16576 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-mulg 18217 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-rnghom 19463 df-psr 20594 df-mpl 20596 |
This theorem is referenced by: evlslem1 20754 |
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