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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 20862 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → 𝐵 ∈ (SubRing‘𝑆)) |
8 | subrgsubg 19534 | . . . . 5 ⊢ (𝐵 ∈ (SubRing‘𝑆) → 𝐵 ∈ (SubGrp‘𝑆)) | |
9 | subgsubm 18293 | . . . . 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 2798 | . . 3 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s 𝐵) | |
14 | 1, 11, 12, 13 | gsumsubm 17991 | . 2 ⊢ (𝜑 → (𝑆 Σg 𝐹) = ((𝑆 ↾s 𝐵) Σg 𝐹)) |
15 | fex 6966 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
16 | 12, 1, 15 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
17 | ovexd 7170 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ V) | |
18 | 5 | fvexi 6659 | . . . 4 ⊢ 𝑈 ∈ V |
19 | 18 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑈 ∈ V) |
20 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
21 | 6 | oveq2i 7146 | . . . . 5 ⊢ (𝑆 ↾s 𝐵) = (𝑆 ↾s (Base‘𝑈)) |
22 | 3, 4, 5, 20, 2, 21 | ressply1bas 20858 | . . . 4 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(𝑆 ↾s 𝐵))) |
23 | 22 | eqcomd 2804 | . . 3 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = (Base‘𝑈)) |
24 | 13 | subrgring 19531 | . . . . 5 ⊢ (𝐵 ∈ (SubRing‘𝑆) → (𝑆 ↾s 𝐵) ∈ Ring) |
25 | 7, 24 | syl 17 | . . . 4 ⊢ (𝑇 ∈ (SubRing‘𝑅) → (𝑆 ↾s 𝐵) ∈ Ring) |
26 | ringmgm 19301 | . . . 4 ⊢ ((𝑆 ↾s 𝐵) ∈ Ring → (𝑆 ↾s 𝐵) ∈ Mgm) | |
27 | 2, 25, 26 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝑆 ↾s 𝐵) ∈ Mgm) |
28 | simpl 486 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝜑) | |
29 | 3, 4, 5, 6, 2, 13 | ressply1bas 20858 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆 ↾s 𝐵))) |
30 | 29 | eqcomd 2804 | . . . . . . . . 9 ⊢ (𝜑 → (Base‘(𝑆 ↾s 𝐵)) = 𝐵) |
31 | 30 | eleq2d 2875 | . . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑠 ∈ 𝐵)) |
32 | 31 | biimpcd 252 | . . . . . . 7 ⊢ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑠 ∈ 𝐵)) |
33 | 32 | adantr 484 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑠 ∈ 𝐵)) |
34 | 33 | impcom 411 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑠 ∈ 𝐵) |
35 | 30 | eleq2d 2875 | . . . . . . . 8 ⊢ (𝜑 → (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) ↔ 𝑡 ∈ 𝐵)) |
36 | 35 | biimpcd 252 | . . . . . . 7 ⊢ (𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)) → (𝜑 → 𝑡 ∈ 𝐵)) |
37 | 36 | adantl 485 | . . . . . 6 ⊢ ((𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵))) → (𝜑 → 𝑡 ∈ 𝐵)) |
38 | 37 | impcom 411 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘(𝑆 ↾s 𝐵)) ∧ 𝑡 ∈ (Base‘(𝑆 ↾s 𝐵)))) → 𝑡 ∈ 𝐵) |
39 | 3, 4, 5, 6, 2, 13 | ressply1add 20859 | . . . . 5 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑠(+g‘𝑈)𝑡) = (𝑠(+g‘(𝑆 ↾s 𝐵))𝑡)) |
40 | 28, 34, 38, 39 | syl12anc 835 | . . . 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 2804 | . . 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 6491 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
43 | 12 | frnd 6494 | . . . 4 ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
44 | 43, 29 | sseqtrd 3955 | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘(𝑆 ↾s 𝐵))) |
45 | 16, 17, 19, 23, 27, 41, 42, 44 | gsummgmpropd 17883 | . 2 ⊢ (𝜑 → ((𝑆 ↾s 𝐵) Σg 𝐹) = (𝑈 Σg 𝐹)) |
46 | 14, 45 | eqtrd 2833 | 1 ⊢ (𝜑 → (𝑆 Σg 𝐹) = (𝑈 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 Σg cgsu 16706 Mgmcmgm 17842 SubMndcsubmnd 17947 SubGrpcsubg 18265 Ringcrg 19290 SubRingcsubrg 19524 Poly1cpl1 20806 |
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-iin 4884 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-ofr 7390 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-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 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-dec 12087 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-ple 16577 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-subrg 19526 df-psr 20594 df-mpl 20596 df-opsr 20598 df-psr1 20809 df-ply1 20811 |
This theorem is referenced by: (None) |
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