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Mirrors > Home > MPE Home > Th. List > frlmsubgval | Structured version Visualization version GIF version |
Description: Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
Ref | Expression |
---|---|
frlmsubval.y | ⊢ 𝑌 = (𝑅 freeLMod 𝐼) |
frlmsubval.b | ⊢ 𝐵 = (Base‘𝑌) |
frlmsubval.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
frlmsubval.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
frlmsubval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
frlmsubval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
frlmsubval.a | ⊢ − = (-g‘𝑅) |
frlmsubval.p | ⊢ 𝑀 = (-g‘𝑌) |
Ref | Expression |
---|---|
frlmsubgval | ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘f − 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmsubval.p | . . . 4 ⊢ 𝑀 = (-g‘𝑌) | |
2 | frlmsubval.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
3 | frlmsubval.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
4 | frlmsubval.y | . . . . . . 7 ⊢ 𝑌 = (𝑅 freeLMod 𝐼) | |
5 | frlmsubval.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑌) | |
6 | 4, 5 | frlmpws 20684 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
7 | 2, 3, 6 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 7 | fveq2d 6710 | . . . 4 ⊢ (𝜑 → (-g‘𝑌) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
9 | 1, 8 | syl5eq 2786 | . . 3 ⊢ (𝜑 → 𝑀 = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | 9 | oveqd 7219 | . 2 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
11 | rlmlmod 20214 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
12 | 2, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
13 | eqid 2734 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
14 | 13 | pwslmod 19979 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
15 | 12, 3, 14 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
16 | eqid 2734 | . . . . . 6 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
17 | 4, 5, 16 | frlmlss 20685 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
18 | 2, 3, 17 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
19 | 16 | lsssubg 19966 | . . . 4 ⊢ ((((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
20 | 15, 18, 19 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
21 | frlmsubval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
22 | frlmsubval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
23 | eqid 2734 | . . . 4 ⊢ (-g‘((ringLMod‘𝑅) ↑s 𝐼)) = (-g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
24 | eqid 2734 | . . . 4 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | eqid 2734 | . . . 4 ⊢ (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) | |
26 | 23, 24, 25 | subgsub 18527 | . . 3 ⊢ ((𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼)) ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
27 | 20, 21, 22, 26 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
28 | lmodgrp 19878 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (ringLMod‘𝑅) ∈ Grp) | |
29 | 2, 11, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ Grp) |
30 | eqid 2734 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
31 | 4, 30, 5 | frlmbasmap 20693 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ ((Base‘𝑅) ↑m 𝐼)) |
32 | 3, 21, 31 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑅) ↑m 𝐼)) |
33 | rlmbas 20204 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
34 | 13, 33 | pwsbas 16964 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ Grp ∧ 𝐼 ∈ 𝑊) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
35 | 29, 3, 34 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑m 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
36 | 32, 35 | eleqtrd 2836 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 4, 30, 5 | frlmbasmap 20693 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ ((Base‘𝑅) ↑m 𝐼)) |
38 | 3, 22, 37 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑅) ↑m 𝐼)) |
39 | 38, 35 | eleqtrd 2836 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
40 | eqid 2734 | . . . 4 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
41 | frlmsubval.a | . . . . 5 ⊢ − = (-g‘𝑅) | |
42 | rlmsub 20207 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅)) | |
43 | 41, 42 | eqtri 2762 | . . . 4 ⊢ − = (-g‘(ringLMod‘𝑅)) |
44 | 13, 40, 43, 23 | pwssub 18449 | . . 3 ⊢ ((((ringLMod‘𝑅) ∈ Grp ∧ 𝐼 ∈ 𝑊) ∧ (𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) ∧ 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼)))) → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘f − 𝐺)) |
45 | 29, 3, 36, 39, 44 | syl22anc 839 | . 2 ⊢ (𝜑 → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘f − 𝐺)) |
46 | 10, 27, 45 | 3eqtr2d 2780 | 1 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘f − 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6369 (class class class)co 7202 ∘f cof 7456 ↑m cmap 8497 Basecbs 16684 ↾s cress 16685 ↑s cpws 16923 Grpcgrp 18337 -gcsg 18339 SubGrpcsubg 18509 Ringcrg 19534 LModclmod 19871 LSubSpclss 19940 ringLModcrglmod 20178 freeLMod cfrlm 20680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-sup 9047 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-hom 16791 df-cco 16792 df-0g 16918 df-prds 16924 df-pws 16926 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-sbg 18342 df-subg 18512 df-mgp 19477 df-ur 19489 df-ring 19536 df-subrg 19770 df-lmod 19873 df-lss 19941 df-sra 20181 df-rgmod 20182 df-dsmm 20666 df-frlm 20681 |
This theorem is referenced by: matsubgcell 21303 rrxds 24262 |
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