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Mirrors > Home > MPE Home > Th. List > rspval | Structured version Visualization version GIF version |
Description: Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
Ref | Expression |
---|---|
rspval | ⊢ (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rsp 19876 | . . 3 ⊢ RSpan = (LSpan ∘ ringLMod) | |
2 | 1 | fveq1i 6664 | . 2 ⊢ (RSpan‘𝑊) = ((LSpan ∘ ringLMod)‘𝑊) |
3 | 00lsp 19682 | . . 3 ⊢ ∅ = (LSpan‘∅) | |
4 | rlmfn 19891 | . . . 4 ⊢ ringLMod Fn V | |
5 | fnfun 6446 | . . . 4 ⊢ (ringLMod Fn V → Fun ringLMod) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ Fun ringLMod |
7 | 3, 6 | fvco4i 6755 | . 2 ⊢ ((LSpan ∘ ringLMod)‘𝑊) = (LSpan‘(ringLMod‘𝑊)) |
8 | 2, 7 | eqtri 2841 | 1 ⊢ (RSpan‘𝑊) = (LSpan‘(ringLMod‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Vcvv 3492 ∘ ccom 5552 Fun wfun 6342 Fn wfn 6343 ‘cfv 6348 LSpanclspn 19672 ringLModcrglmod 19870 RSpancrsp 19872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-slot 16475 df-base 16477 df-lss 19633 df-lsp 19673 df-rgmod 19874 df-rsp 19876 |
This theorem is referenced by: rspcl 19923 rspssid 19924 rsp0 19926 rspssp 19927 mrcrsp 19928 lidlrsppropd 19931 rspsn 19955 rspsnel 30863 rgmoddim 30907 islnr2 39592 |
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