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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 20459). (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 19567 | . . . . 5 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
2 | eqid 2826 | . . . . . 6 ⊢ ((ringLMod‘𝑅) ↑s 𝐼) = ((ringLMod‘𝑅) ↑s 𝐼) | |
3 | 2 | pwslmod 19330 | . . . . 5 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
4 | 1, 3 | sylan 577 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → ((ringLMod‘𝑅) ↑s 𝐼) ∈ LMod) |
5 | frlmval.f | . . . . 5 ⊢ 𝐹 = (𝑅 freeLMod 𝐼) | |
6 | eqid 2826 | . . . . 5 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2826 | . . . . 5 ⊢ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼)) | |
8 | 5, 6, 7 | frlmlss 20459 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) ∈ (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))) |
9 | 7 | lsssubg 19317 | . . . 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 581 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (Base‘𝐹) ∈ (SubGrp‘((ringLMod‘𝑅) ↑s 𝐼))) |
11 | eqid 2826 | . . . 4 ⊢ (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)) = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)) | |
12 | eqid 2826 | . . . 4 ⊢ (0g‘((ringLMod‘𝑅) ↑s 𝐼)) = (0g‘((ringLMod‘𝑅) ↑s 𝐼)) | |
13 | 11, 12 | subg0 17952 | . . 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 19227 | . . . 4 ⊢ ((ringLMod‘𝑅) ∈ LMod → (ringLMod‘𝑅) ∈ Grp) | |
16 | grpmnd 17784 | . . . 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 19559 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘(ringLMod‘𝑅)) | |
20 | 18, 19 | eqtri 2850 | . . . 4 ⊢ 0 = (0g‘(ringLMod‘𝑅)) |
21 | 2, 20 | pws0g 17680 | . . 3 ⊢ (((ringLMod‘𝑅) ∈ Mnd ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘((ringLMod‘𝑅) ↑s 𝐼))) |
22 | 17, 21 | sylan 577 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘((ringLMod‘𝑅) ↑s 𝐼))) |
23 | 5, 6 | frlmpws 20458 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹))) |
24 | 23 | fveq2d 6438 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (0g‘𝐹) = (0g‘(((ringLMod‘𝑅) ↑s 𝐼) ↾s (Base‘𝐹)))) |
25 | 14, 22, 24 | 3eqtr4d 2872 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {csn 4398 × cxp 5341 ‘cfv 6124 (class class class)co 6906 Basecbs 16223 ↾s cress 16224 0gc0g 16454 ↑s cpws 16461 Mndcmnd 17648 Grpcgrp 17777 SubGrpcsubg 17940 Ringcrg 18902 LModclmod 19220 LSubSpclss 19289 ringLModcrglmod 19531 freeLMod cfrlm 20454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-map 8125 df-ixp 8177 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-hom 16330 df-cco 16331 df-0g 16456 df-prds 16462 df-pws 16464 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-mgp 18845 df-ur 18857 df-ring 18904 df-subrg 19135 df-lmod 19222 df-lss 19290 df-sra 19534 df-rgmod 19535 df-dsmm 20440 df-frlm 20455 |
This theorem is referenced by: frlmsslss 20481 islindf5 20546 mat0op 20593 rrxcph 23561 rrx0 23566 matunitlindflem1 33950 zlmodzxz0 42982 aacllem 43444 |
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