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Mirrors > Home > MPE Home > Th. List > rlmscaf | Structured version Visualization version GIF version |
Description: Functionalized scalar multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
rlmscaf | ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . . 4 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | eqid 2823 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | mgpbas 19247 | . . 3 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
4 | eqid 2823 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 4 | mgpplusg 19245 | . . 3 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
6 | eqid 2823 | . . 3 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (+𝑓‘(mulGrp‘𝑅)) | |
7 | 3, 5, 6 | plusffval 17860 | . 2 ⊢ (+𝑓‘(mulGrp‘𝑅)) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (𝑥(.r‘𝑅)𝑦)) |
8 | rlmbas 19969 | . . 3 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
9 | rlmsca2 19975 | . . 3 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
10 | df-base 16491 | . . . 4 ⊢ Base = Slot 1 | |
11 | 10, 2 | strfvi 16539 | . . 3 ⊢ (Base‘𝑅) = (Base‘( I ‘𝑅)) |
12 | eqid 2823 | . . 3 ⊢ ( ·sf ‘(ringLMod‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) | |
13 | rlmvsca 19976 | . . 3 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
14 | 8, 9, 11, 12, 13 | scaffval 19654 | . 2 ⊢ ( ·sf ‘(ringLMod‘𝑅)) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (𝑥(.r‘𝑅)𝑦)) |
15 | 7, 14 | eqtr4i 2849 | 1 ⊢ (+𝑓‘(mulGrp‘𝑅)) = ( ·sf ‘(ringLMod‘𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 I cid 5461 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1c1 10540 Basecbs 16485 .rcmulr 16568 +𝑓cplusf 17851 mulGrpcmgp 19241 ·sf cscaf 19637 ringLModcrglmod 19943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-sca 16583 df-vsca 16584 df-ip 16585 df-plusf 17853 df-mgp 19242 df-scaf 19639 df-sra 19946 df-rgmod 19947 |
This theorem is referenced by: nrgtrg 23301 |
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