![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > frlmval | Structured version Visualization version GIF version |
Description: Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
frlmval.f | β’ πΉ = (π freeLMod πΌ) |
Ref | Expression |
---|---|
frlmval | β’ ((π β π β§ πΌ β π) β πΉ = (π βm (πΌ Γ {(ringLModβπ )}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frlmval.f | . 2 β’ πΉ = (π freeLMod πΌ) | |
2 | elex 3487 | . . 3 β’ (π β π β π β V) | |
3 | elex 3487 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | id 22 | . . . . 5 β’ (π = π β π = π ) | |
5 | fveq2 6884 | . . . . . . 7 β’ (π = π β (ringLModβπ) = (ringLModβπ )) | |
6 | 5 | sneqd 4635 | . . . . . 6 β’ (π = π β {(ringLModβπ)} = {(ringLModβπ )}) |
7 | 6 | xpeq2d 5699 | . . . . 5 β’ (π = π β (π Γ {(ringLModβπ)}) = (π Γ {(ringLModβπ )})) |
8 | 4, 7 | oveq12d 7422 | . . . 4 β’ (π = π β (π βm (π Γ {(ringLModβπ)})) = (π βm (π Γ {(ringLModβπ )}))) |
9 | xpeq1 5683 | . . . . 5 β’ (π = πΌ β (π Γ {(ringLModβπ )}) = (πΌ Γ {(ringLModβπ )})) | |
10 | 9 | oveq2d 7420 | . . . 4 β’ (π = πΌ β (π βm (π Γ {(ringLModβπ )})) = (π βm (πΌ Γ {(ringLModβπ )}))) |
11 | df-frlm 21637 | . . . 4 β’ freeLMod = (π β V, π β V β¦ (π βm (π Γ {(ringLModβπ)}))) | |
12 | ovex 7437 | . . . 4 β’ (π βm (πΌ Γ {(ringLModβπ )})) β V | |
13 | 8, 10, 11, 12 | ovmpo 7563 | . . 3 β’ ((π β V β§ πΌ β V) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
14 | 2, 3, 13 | syl2an 595 | . 2 β’ ((π β π β§ πΌ β π) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
15 | 1, 14 | eqtrid 2778 | 1 β’ ((π β π β§ πΌ β π) β πΉ = (π βm (πΌ Γ {(ringLModβπ )}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 {csn 4623 Γ cxp 5667 βcfv 6536 (class class class)co 7404 ringLModcrglmod 21017 βm cdsmm 21621 freeLMod cfrlm 21636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-frlm 21637 |
This theorem is referenced by: frlmlmod 21639 frlmpws 21640 frlmlss 21641 frlmpwsfi 21642 frlmbas 21645 |
Copyright terms: Public domain | W3C validator |