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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 20403 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) |
8 | subrgsubg 19543 | . . . . 5 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆)) | |
9 | subgsubm 18303 | . . . . 5 ⊢ (𝐵 ∈ (SubGrp‘𝑆) → 𝐵 ∈ (SubMnd‘𝑆)) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubMnd‘𝑆)) |
11 | 2, 7, 10 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubMnd‘𝑆)) |
12 | gsumply1subr.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
13 | eqid 2823 | . . 3 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
14 | 1, 11, 12, 13 | gsumsubm 18001 | . 2 ⊢ (𝜑 → (𝑆 Σg 𝐹) = ((𝑆 ↾s 𝐵) Σg 𝐹)) |
15 | fex 6991 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
16 | 12, 1, 15 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | ovexd 7193 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ V) | |
18 | 5 | fvexi 6686 | . . . 4 ⊢ 𝑈 ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
20 | eqid 2823 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
21 | 6 | oveq2i 7169 | . . . . 5 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s (Base‘𝑈)) |
22 | 3, 4, 5, 20, 2, 21 | ressply1bas 20399 | . . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾s 𝐵))) |
23 | 22 | eqcomd 2829 | . . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = (Base‘𝑈)) |
24 | 13 | subrgring 19540 | . . . . 5 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝐵) ∈ Ring) |
25 | 7, 24 | syl 17 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (𝑆 ↾s 𝐵) ∈ Ring) |
26 | ringmgm 19309 | . . . 4 ⊢ ((𝑆 ↾s 𝐵) ∈ Ring → (𝑆 ↾s 𝐵) ∈ Mgm) | |
27 | 2, 25, 26 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ Mgm) |
28 | simpl 485 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝜑) | |
29 | 3, 4, 5, 6, 2, 13 | ressply1bas 20399 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾s 𝐵))) |
30 | 29 | eqcomd 2829 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = 𝐵) |
31 | 30 | eleq2d 2900 | . . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑠 ∈ 𝐵)) |
32 | 31 | biimpcd 251 | . . . . . . 7 ⊢ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑠 ∈ 𝐵)) |
33 | 32 | adantr 483 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑠 ∈ 𝐵)) |
34 | 33 | impcom 410 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑠 ∈ 𝐵) |
35 | 30 | eleq2d 2900 | . . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑡 ∈ 𝐵)) |
36 | 35 | biimpcd 251 | . . . . . . 7 ⊢ (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑡 ∈ 𝐵)) |
37 | 36 | adantl 484 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑡 ∈ 𝐵)) |
38 | 37 | impcom 410 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑡 ∈ 𝐵) |
39 | 3, 4, 5, 6, 2, 13 | ressply1add 20400 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+g‘𝑈)𝑡) = (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡)) |
40 | 28, 34, 38, 39 | syl12anc 834 | . . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → (𝑠(+g‘𝑈)𝑡) = (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡)) |
41 | 40 | eqcomd 2829 | . . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡) = (𝑠(+g‘𝑈)𝑡)) |
42 | 12 | ffund 6520 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
43 | 12 | frnd 6523 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
44 | 43, 29 | sseqtrd 4009 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾s 𝐵))) |
45 | 16, 17, 19, 23, 27, 41, 42, 44 | gsummgmpropd 17893 | . 2 ⊢ (𝜑 → ((𝑆 ↾s 𝐵) Σg 𝐹) = (𝑈 Σg 𝐹)) |
46 | 14, 45 | eqtrd 2858 | 1 ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 +gcplusg 16567 Σg cgsu 16716 Mgmcmgm 17852 SubMndcsubmnd 17957 SubGrpcsubg 18275 Ringcrg 19299 SubRingcsubrg 19533 Poly1cpl1 20347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-tset 16586 df-ple 16587 df-0g 16717 df-gsum 16718 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-ghm 18358 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-subrg 19535 df-psr 20138 df-mpl 20140 df-opsr 20142 df-psr1 20350 df-ply1 20352 |
This theorem is referenced by: (None) |
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