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| Mirrors > Home > MPE Home > Th. List > gsumply1subr | Structured version Visualization version GIF version | ||
| Description: Evaluate a group sum in a polynomial ring over a subring. (Contributed by AV, 22-Sep-2019.) (Proof shortened by AV, 31-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| subrgply1.s | ⊢ 𝑆 = (Poly1‘𝑅) | 
| subrgply1.h | ⊢ 𝐻 = (𝑅 ↾s 𝑇) | 
| subrgply1.u | ⊢ 𝑈 = (Poly1‘𝐻) | 
| subrgply1.b | ⊢ 𝐵 = (Base‘𝑈) | 
| gsumply1subr.s | ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | 
| gsumply1subr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) | 
| gsumply1subr.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| Ref | Expression | 
|---|---|
| gsumply1subr | ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | gsumply1subr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 2 | gsumply1subr.s | . . . 4 ⊢ (𝜑 → 𝑇 ∈ (SubRing‘𝑅)) | |
| 3 | subrgply1.s | . . . . 5 ⊢ 𝑆 = (Poly1‘𝑅) | |
| 4 | subrgply1.h | . . . . 5 ⊢ 𝐻 = (𝑅 ↾s 𝑇) | |
| 5 | subrgply1.u | . . . . 5 ⊢ 𝑈 = (Poly1‘𝐻) | |
| 6 | subrgply1.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑈) | |
| 7 | 3, 4, 5, 6 | subrgply1 22235 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) | 
| 8 | subrgsubg 20578 | . . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆)) | |
| 9 | subgsubm 19167 | . . . 4 ⊢ (𝐵 ∈ (SubGrp‘𝑆) → 𝐵 ∈ (SubMnd‘𝑆)) | |
| 10 | 2, 7, 8, 9 | 4syl 19 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝑆)) | 
| 11 | gsumply1subr.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 12 | eqid 2736 | . . 3 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
| 13 | 1, 10, 11, 12 | gsumsubm 18849 | . 2 ⊢ (𝜑 → (𝑆 Σg 𝐹) = ((𝑆 ↾s 𝐵) Σg 𝐹)) | 
| 14 | 11, 1 | fexd 7248 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) | 
| 15 | ovexd 7467 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ V) | |
| 16 | 5 | fvexi 6919 | . . . 4 ⊢ 𝑈 ∈ V | 
| 17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) | 
| 18 | eqid 2736 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
| 19 | 6 | oveq2i 7443 | . . . . 5 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s (Base‘𝑈)) | 
| 20 | 3, 4, 5, 18, 2, 19 | ressply1bas 22231 | . . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾s 𝐵))) | 
| 21 | 20 | eqcomd 2742 | . . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = (Base‘𝑈)) | 
| 22 | 12 | subrgring 20575 | . . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝐵) ∈ Ring) | 
| 23 | ringmgm 20242 | . . . 4 ⊢ ((𝑆 ↾s 𝐵) ∈ Ring → (𝑆 ↾s 𝐵) ∈ Mgm) | |
| 24 | 2, 7, 22, 23 | 4syl 19 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ Mgm) | 
| 25 | simpl 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝜑) | |
| 26 | 3, 4, 5, 6, 2, 12 | ressply1bas 22231 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾s 𝐵))) | 
| 27 | 26 | eqcomd 2742 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = 𝐵) | 
| 28 | 27 | eleq2d 2826 | . . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑠 ∈ 𝐵)) | 
| 29 | 28 | biimpcd 249 | . . . . . . 7 ⊢ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑠 ∈ 𝐵)) | 
| 30 | 29 | adantr 480 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑠 ∈ 𝐵)) | 
| 31 | 30 | impcom 407 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑠 ∈ 𝐵) | 
| 32 | 27 | eleq2d 2826 | . . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑡 ∈ 𝐵)) | 
| 33 | 32 | biimpcd 249 | . . . . . . 7 ⊢ (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑡 ∈ 𝐵)) | 
| 34 | 33 | adantl 481 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑡 ∈ 𝐵)) | 
| 35 | 34 | impcom 407 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑡 ∈ 𝐵) | 
| 36 | 3, 4, 5, 6, 2, 12 | ressply1add 22232 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+g‘𝑈)𝑡) = (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡)) | 
| 37 | 25, 31, 35, 36 | syl12anc 836 | . . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → (𝑠(+g‘𝑈)𝑡) = (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡)) | 
| 38 | 37 | eqcomd 2742 | . . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡) = (𝑠(+g‘𝑈)𝑡)) | 
| 39 | 11 | ffund 6739 | . . 3 ⊢ (𝜑 → Fun 𝐹) | 
| 40 | 11 | frnd 6743 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) | 
| 41 | 40, 26 | sseqtrd 4019 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾s 𝐵))) | 
| 42 | 14, 15, 17, 21, 24, 38, 39, 41 | gsummgmpropd 18695 | . 2 ⊢ (𝜑 → ((𝑆 ↾s 𝐵) Σg 𝐹) = (𝑈 Σg 𝐹)) | 
| 43 | 13, 42 | eqtrd 2776 | 1 ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ran crn 5685 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 ↾s cress 17275 +gcplusg 17298 Σg cgsu 17486 Mgmcmgm 18652 SubMndcsubmnd 18796 SubGrpcsubg 19139 Ringcrg 20231 SubRingcsubrg 20570 Poly1cpl1 22179 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-ofr 7699 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-subrng 20547 df-subrg 20571 df-psr 21930 df-mpl 21932 df-opsr 21934 df-psr1 22182 df-ply1 22184 | 
| This theorem is referenced by: (None) | 
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