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| Description: Lemma for evlseu 22107. 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 20242 | . . . . . 6 ⊢ (𝑆 ∈ CRing → 𝑆 ∈ Ring) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring) | 
| 4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑆 ∈ Ring) | 
| 5 | evlslem1.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑅 RingHom 𝑆)) | |
| 6 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 7 | evlslem1.c | . . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆) | |
| 8 | 6, 7 | rhmf 20485 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐶) | 
| 9 | 5, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶𝐶) | 
| 10 | 9 | adantr 480 | . . . . 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 22018 | . . . . . 6 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅)) | 
| 16 | 15 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ (Base‘𝑅)) | 
| 17 | 10, 16 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ 𝐶) | 
| 18 | evlslem1.t | . . . . . 6 ⊢ 𝑇 = (mulGrp‘𝑆) | |
| 19 | 18, 7 | mgpbas 20142 | . . . . 5 ⊢ 𝐶 = (Base‘𝑇) | 
| 20 | evlslem1.x | . . . . 5 ⊢ ↑ = (.g‘𝑇) | |
| 21 | 18 | crngmgp 20238 | . . . . . . 7 ⊢ (𝑆 ∈ CRing → 𝑇 ∈ CMnd) | 
| 22 | 1, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ CMnd) | 
| 23 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑇 ∈ CMnd) | 
| 24 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝑏 ∈ 𝐷) | |
| 25 | evlslem1.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐼⟶𝐶) | |
| 26 | 25 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → 𝐺:𝐼⟶𝐶) | 
| 27 | 13, 19, 20, 23, 24, 26 | psrbagev2 22102 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑇 Σg (𝑏 ∘f ↑ 𝐺)) ∈ 𝐶) | 
| 28 | evlslem1.m | . . . . 5 ⊢ · = (.r‘𝑆) | |
| 29 | 7, 28 | ringcl 20247 | . . . 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 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺))) ∈ 𝐶) | 
| 31 | 30 | fmpttd 7135 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))):𝐷⟶𝐶) | 
| 32 | ovexd 7466 | . . . . 5 ⊢ (𝜑 → (ℕ0 ↑m 𝐼) ∈ V) | |
| 33 | 13, 32 | rabexd 5340 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ V) | 
| 34 | 33 | mptexd 7244 | . . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) ∈ V) | 
| 35 | funmpt 6604 | . . . 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 6921 | . . 3 ⊢ (𝜑 → (0g‘𝑆) ∈ V) | |
| 38 | eqid 2737 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
| 39 | 11, 12, 38, 14 | mplelsfi 22015 | . . . 4 ⊢ (𝜑 → 𝑌 finSupp (0g‘𝑅)) | 
| 40 | 39 | fsuppimpd 9409 | . . 3 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) ∈ Fin) | 
| 41 | 15 | feqmptd 6977 | . . . . . . 7 ⊢ (𝜑 → 𝑌 = (𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏))) | 
| 42 | 41 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → (𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅))) | 
| 43 | eqimss2 4043 | . . . . . 6 ⊢ ((𝑌 supp (0g‘𝑅)) = ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) | |
| 44 | 42, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝑌‘𝑏)) supp (0g‘𝑅)) ⊆ (𝑌 supp (0g‘𝑅))) | 
| 45 | rhmghm 20484 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
| 46 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
| 47 | 38, 46 | ghmid 19240 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) | 
| 48 | 5, 45, 47 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = (0g‘𝑆)) | 
| 49 | fvexd 6921 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝑌‘𝑏) ∈ V) | |
| 50 | fvexd 6921 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
| 51 | 44, 48, 49, 50 | suppssfv 8227 | . . . 4 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ (𝐹‘(𝑌‘𝑏))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) | 
| 52 | 7, 28, 46 | ringlz 20290 | . . . . 5 ⊢ ((𝑆 ∈ Ring ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) | 
| 53 | 3, 52 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((0g‘𝑆) · 𝑥) = (0g‘𝑆)) | 
| 54 | fvexd 6921 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐷) → (𝐹‘(𝑌‘𝑏)) ∈ V) | |
| 55 | 51, 53, 54, 27, 37 | suppssov1 8222 | . . 3 ⊢ (𝜑 → ((𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) supp (0g‘𝑆)) ⊆ (𝑌 supp (0g‘𝑅))) | 
| 56 | suppssfifsupp 9420 | . . 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)) | |
| 57 | 34, 36, 37, 40, 55, 56 | syl32anc 1380 | . 2 ⊢ (𝜑 → (𝑏 ∈ 𝐷 ↦ ((𝐹‘(𝑌‘𝑏)) · (𝑇 Σg (𝑏 ∘f ↑ 𝐺)))) finSupp (0g‘𝑆)) | 
| 58 | 31, 57 | jca 511 | 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 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∘f cof 7695 supp csupp 8185 ↑m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 .rcmulr 17298 0gc0g 17484 Σg cgsu 17485 .gcmg 19085 GrpHom cghm 19230 CMndccmn 19798 mulGrpcmgp 20137 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 mVar cmvr 21925 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-mhm 18796 df-grp 18954 df-minusg 18955 df-mulg 19086 df-ghm 19231 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-rhm 20472 df-psr 21929 df-mpl 21931 | 
| This theorem is referenced by: evlslem1 22106 | 
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