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Mirrors > Home > MPE Home > Th. List > frlm0 | Structured version Visualization version GIF version |
Description: Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21685). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | ⊢ 𝐹 = (𝑅 freeLMod 𝐼) |
frlm0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
frlm0 | ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlmlmod 21096 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
2 | eqid 2728 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
3 | 2 | pwslmod 20854 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
4 | 1, 3 | sylan 579 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
5 | frlmval.f | . . . . 5 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
6 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
8 | 5, 6, 7 | frlmlss 21685 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
9 | 7 | lsssubg 20841 | . . . 4 ⊢ ((((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod ∧ (Base‘𝐹) ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) → (Base‘𝐹) ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
10 | 4, 8, 9 | syl2anc 583 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
11 | eqid 2728 | . . . 4 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)) | |
12 | eqid 2728 | . . . 4 ⊢ (0g‘((ringLMod‘𝑅) ↑s 𝐼)) = (0g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
13 | 11, 12 | subg0 19087 | . . 3 ⊢ ((Base‘𝐹) ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼)) → (0g‘((ringLMod‘𝑅) ↑s 𝐼)) = (0g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)))) |
14 | 10, 13 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (0g‘((ringLMod‘𝑅) ↑s 𝐼)) = (0g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)))) |
15 | lmodgrp 20750 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (ringLMod‘𝑅) ∈ Grp) | |
16 | grpmnd 18897 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ Grp → (ringLMod‘𝑅) ∈ Mnd) | |
17 | 1, 15, 16 | 3syl 18 | . . 3 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ Mnd) |
18 | frlm0.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
19 | rlm0 21088 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | |
20 | 18, 19 | eqtri 2756 | . . . 4 ⊢ 0 = (0g‘(ringLMod‘𝑅)) |
21 | 2, 20 | pws0g 18730 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘((ringLMod‘𝑅) ↑s 𝐼))) |
22 | 17, 21 | sylan 579 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘((ringLMod‘𝑅) ↑s 𝐼))) |
23 | 5, 6 | frlmpws 21684 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹))) |
24 | 23 | fveq2d 6901 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (0g‘𝐹) = (0g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)))) |
25 | 14, 22, 24 | 3eqtr4d 2778 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {csn 4629 × cxp 5676 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 ↾s cress 17209 0gc0g 17421 ↑s cpws 17428 Mndcmnd 18694 Grpcgrp 18890 SubGrpcsubg 19075 Ringcrg 20173 LModclmod 20743 LSubSpclss 20815 ringLModcrglmod 21057 freeLMod cfrlm 21680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-subrg 20508 df-lmod 20745 df-lss 20816 df-sra 21058 df-rgmod 21059 df-dsmm 21666 df-frlm 21681 |
This theorem is referenced by: frlmsslss 21708 islindf5 21773 mat0op 22334 rrxcph 25333 rrx0 25338 matunitlindflem1 37089 frlm0vald 41769 mnring0g2d 43657 zlmodzxz0 47420 aacllem 48234 |
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