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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 21788 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
7 | 2, 3, 6 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) |
8 | 7 | fveq2d 6911 | . . . 4 ⊢ (𝜑 → (-g‘𝑌) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
9 | 1, 8 | eqtrid 2787 | . . 3 ⊢ (𝜑 → 𝑀 = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))) |
10 | 9 | oveqd 7448 | . 2 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
11 | rlmlmod 21228 | . . . . . 6 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
12 | 2, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ LMod) |
13 | eqid 2735 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
14 | 13 | pwslmod 20986 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
15 | 12, 3, 14 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
16 | eqid 2735 | . . . . . 6 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
17 | 4, 5, 16 | frlmlss 21789 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
18 | 2, 3, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
19 | 16 | lsssubg 20973 | . . . 4 ⊢ ((((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod ∧ 𝐵 ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
20 | 15, 18, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
21 | frlmsubval.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
22 | frlmsubval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
23 | eqid 2735 | . . . 4 ⊢ (-g‘((ringLMod‘𝑅) ↑s 𝐼)) = (-g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
24 | eqid 2735 | . . . 4 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵) | |
25 | eqid 2735 | . . . 4 ⊢ (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) = (-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) | |
26 | 23, 24, 25 | subgsub 19169 | . . 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 1370 | . 2 ⊢ (𝜑 → (𝐹(-g‘((ringLMod‘𝑅) ↑s 𝐼))𝐺) = (𝐹(-g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))𝐺)) |
28 | lmodgrp 20882 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (ringLMod‘𝑅) ∈ Grp) | |
29 | 2, 11, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → (ringLMod‘𝑅) ∈ Grp) |
30 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
31 | 4, 30, 5 | frlmbasmap 21797 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐹 ∈ 𝐵) → 𝐹 ∈ ((Base‘𝑅) ↑m 𝐼)) |
32 | 3, 21, 31 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ((Base‘𝑅) ↑m 𝐼)) |
33 | rlmbas 21218 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
34 | 13, 33 | pwsbas 17534 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ Grp ∧ 𝐼 ∈ 𝑊) → ((Base‘𝑅) ↑m 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
35 | 29, 3, 34 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((Base‘𝑅) ↑m 𝐼) = (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
36 | 32, 35 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
37 | 4, 30, 5 | frlmbasmap 21797 | . . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ ((Base‘𝑅) ↑m 𝐼)) |
38 | 3, 22, 37 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ((Base‘𝑅) ↑m 𝐼)) |
39 | 38, 35 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (Base‘((ringLMod‘𝑅) ↑s 𝐼))) |
40 | eqid 2735 | . . . 4 ⊢ (Base‘((ringLMod‘𝑅) ↑s 𝐼)) = (Base‘((ringLMod‘𝑅) ↑s 𝐼)) | |
41 | frlmsubval.a | . . . . 5 ⊢ − = (-g‘𝑅) | |
42 | rlmsub 21221 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘(ringLMod‘𝑅)) | |
43 | 41, 42 | eqtri 2763 | . . . 4 ⊢ − = (-g‘(ringLMod‘𝑅)) |
44 | 13, 40, 43, 23 | pwssub 19085 | . . 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 2781 | 1 ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘f − 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ∘f cof 7695 ↑m cmap 8865 Basecbs 17245 ↾s cress 17274 ↑s cpws 17493 Grpcgrp 18964 -gcsg 18966 SubGrpcsubg 19151 Ringcrg 20251 LModclmod 20875 LSubSpclss 20947 ringLModcrglmod 21189 freeLMod cfrlm 21784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 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 17488 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrg 20587 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 |
This theorem is referenced by: matsubgcell 22456 rrxds 25441 |
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