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Mirrors > Home > ILE Home > Th. List > rlmfn | GIF version |
Description: ringLMod is a function. (Contributed by Stefan O'Rear, 6-Dec-2014.) |
Ref | Expression |
---|---|
rlmfn | ⊢ ringLMod Fn V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2194 | . . . 4 ⊢ (⊤ → ((subringAlg ‘𝑎)‘(Base‘𝑎)) = ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
2 | ssidd 3201 | . . . 4 ⊢ (⊤ → (Base‘𝑎) ⊆ (Base‘𝑎)) | |
3 | vex 2763 | . . . . 5 ⊢ 𝑎 ∈ V | |
4 | 3 | a1i 9 | . . . 4 ⊢ (⊤ → 𝑎 ∈ V) |
5 | 1, 2, 4 | sraex 13945 | . . 3 ⊢ (⊤ → ((subringAlg ‘𝑎)‘(Base‘𝑎)) ∈ V) |
6 | 5 | mptru 1373 | . 2 ⊢ ((subringAlg ‘𝑎)‘(Base‘𝑎)) ∈ V |
7 | df-rgmod 13935 | . 2 ⊢ ringLMod = (𝑎 ∈ V ↦ ((subringAlg ‘𝑎)‘(Base‘𝑎))) | |
8 | 6, 7 | fnmpti 5383 | 1 ⊢ ringLMod Fn V |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1365 ∈ wcel 2164 Vcvv 2760 Fn wfn 5250 ‘cfv 5255 Basecbs 12621 subringAlg csra 13932 ringLModcrglmod 13933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1re 7968 ax-addrcl 7971 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-7 9048 df-8 9049 df-ndx 12624 df-slot 12625 df-base 12627 df-sets 12628 df-iress 12629 df-mulr 12712 df-sca 12714 df-vsca 12715 df-ip 12716 df-sra 13934 df-rgmod 13935 |
This theorem is referenced by: rlmsubg 13957 rlmvnegg 13964 ixpsnbasval 13965 lidlvalg 13970 rspvalg 13971 lidlex 13972 rspex 13973 lidlmex 13974 lidlss 13975 lidlrsppropdg 13994 |
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