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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 20504 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
7 | 2, 3, 6 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 7 | fveq2d 6452 | . . . 4 ⊢ (𝜑 → (-g‘𝑌) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
9 | 1, 8 | syl5eq 2826 | . . 3 ⊢ (𝜑 → 𝑀 = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | 9 | oveqd 6941 | . 2 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
11 | rlmlmod 19613 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
12 | 2, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
13 | eqid 2778 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
14 | 13 | pwslmod 19376 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
15 | 12, 3, 14 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
16 | eqid 2778 | . . . . . 6 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
17 | 4, 5, 16 | frlmlss 20505 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
18 | 2, 3, 17 | syl2anc 579 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
19 | 16 | lsssubg 19363 | . . . 4 ⊢ ((((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
20 | 15, 18, 19 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
21 | frlmsubval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
22 | frlmsubval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
23 | eqid 2778 | . . . 4 ⊢ (-g‘((ringLMod‘𝑅) ↑s 𝐼)) = (-g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
24 | eqid 2778 | . . . 4 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | eqid 2778 | . . . 4 ⊢ (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) | |
26 | 23, 24, 25 | subgsub 18001 | . . 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 1439 | . 2 ⊢ (𝜑 → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
28 | lmodgrp 19273 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (ringLMod‘𝑅) ∈ Grp) | |
29 | 2, 11, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ Grp) |
30 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
31 | 4, 30, 5 | frlmbasmap 20513 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
32 | 3, 21, 31 | syl2anc 579 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
33 | rlmbas 19603 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
34 | 13, 33 | pwsbas 16544 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ Grp ∧ 𝐼 ∈ 𝑊) → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
35 | 29, 3, 34 | syl2anc 579 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑𝑚 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
36 | 32, 35 | eleqtrd 2861 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 4, 30, 5 | frlmbasmap 20513 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
38 | 3, 22, 37 | syl2anc 579 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑅) ↑𝑚 𝐼)) |
39 | 38, 35 | eleqtrd 2861 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
40 | eqid 2778 | . . . 4 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
41 | frlmsubval.a | . . . . 5 ⊢ − = (-g‘𝑅) | |
42 | rlmsub 19606 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅)) | |
43 | 41, 42 | eqtri 2802 | . . . 4 ⊢ − = (-g‘(ringLMod‘𝑅)) |
44 | 13, 40, 43, 23 | pwssub 17927 | . . 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 829 | . 2 ⊢ (𝜑 → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹 ∘𝑓 − 𝐺)) |
46 | 10, 27, 45 | 3eqtr2d 2820 | 1 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘𝑓 − 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ∘𝑓 cof 7174 ↑𝑚 cmap 8142 Basecbs 16266 ↾s cress 16267 ↑s cpws 16504 Grpcgrp 17820 -gcsg 17822 SubGrpcsubg 17983 Ringcrg 18945 LModclmod 19266 LSubSpclss 19335 ringLModcrglmod 19577 freeLMod cfrlm 20500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-fz 12649 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-sca 16365 df-vsca 16366 df-ip 16367 df-tset 16368 df-ple 16369 df-ds 16371 df-hom 16373 df-cco 16374 df-0g 16499 df-prds 16505 df-pws 16507 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-grp 17823 df-minusg 17824 df-sbg 17825 df-subg 17986 df-mgp 18888 df-ur 18900 df-ring 18947 df-subrg 19181 df-lmod 19268 df-lss 19336 df-sra 19580 df-rgmod 19581 df-dsmm 20486 df-frlm 20501 |
This theorem is referenced by: matsubgcell 20655 rrxds 23610 |
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