![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sralmod0 | Structured version Visualization version GIF version |
Description: The subring module inherits a zero from its ring. (Contributed by Stefan O'Rear, 27-Dec-2014.) |
Ref | Expression |
---|---|
sralmod0.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
sralmod0.z | ⊢ (𝜑 → 0 = (0g‘𝑊)) |
sralmod0.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
sralmod0 | ⊢ (𝜑 → 0 = (0g‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sralmod0.z | . 2 ⊢ (𝜑 → 0 = (0g‘𝑊)) | |
2 | eqidd 2780 | . . 3 ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝑊)) | |
3 | sralmod0.a | . . . 4 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
4 | sralmod0.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
5 | 3, 4 | srabase 19672 | . . 3 ⊢ (𝜑 → (Base‘𝑊) = (Base‘𝐴)) |
6 | 3, 4 | sraaddg 19673 | . . . 4 ⊢ (𝜑 → (+g‘𝑊) = (+g‘𝐴)) |
7 | 6 | oveqdr 7004 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ (Base‘𝑊) ∧ 𝑏 ∈ (Base‘𝑊))) → (𝑎(+g‘𝑊)𝑏) = (𝑎(+g‘𝐴)𝑏)) |
8 | 2, 5, 7 | grpidpropd 17729 | . 2 ⊢ (𝜑 → (0g‘𝑊) = (0g‘𝐴)) |
9 | 1, 8 | eqtrd 2815 | 1 ⊢ (𝜑 → 0 = (0g‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3830 ‘cfv 6188 Basecbs 16339 +gcplusg 16421 0gc0g 16569 subringAlg csra 19662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-plusg 16434 df-sca 16437 df-vsca 16438 df-ip 16439 df-0g 16571 df-sra 19666 |
This theorem is referenced by: rlm0 19691 sradrng 30614 drgext0g 30618 extdg1id 30679 ccfldextdgrr 30683 |
Copyright terms: Public domain | W3C validator |