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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 3492 | . . 3 β’ (π β π β π β V) | |
3 | elex 3492 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | id 22 | . . . . 5 β’ (π = π β π = π ) | |
5 | fveq2 6902 | . . . . . . 7 β’ (π = π β (ringLModβπ) = (ringLModβπ )) | |
6 | 5 | sneqd 4644 | . . . . . 6 β’ (π = π β {(ringLModβπ)} = {(ringLModβπ )}) |
7 | 6 | xpeq2d 5712 | . . . . 5 β’ (π = π β (π Γ {(ringLModβπ)}) = (π Γ {(ringLModβπ )})) |
8 | 4, 7 | oveq12d 7444 | . . . 4 β’ (π = π β (π βm (π Γ {(ringLModβπ)})) = (π βm (π Γ {(ringLModβπ )}))) |
9 | xpeq1 5696 | . . . . 5 β’ (π = πΌ β (π Γ {(ringLModβπ )}) = (πΌ Γ {(ringLModβπ )})) | |
10 | 9 | oveq2d 7442 | . . . 4 β’ (π = πΌ β (π βm (π Γ {(ringLModβπ )})) = (π βm (πΌ Γ {(ringLModβπ )}))) |
11 | df-frlm 21688 | . . . 4 β’ freeLMod = (π β V, π β V β¦ (π βm (π Γ {(ringLModβπ)}))) | |
12 | ovex 7459 | . . . 4 β’ (π βm (πΌ Γ {(ringLModβπ )})) β V | |
13 | 8, 10, 11, 12 | ovmpo 7587 | . . 3 β’ ((π β V β§ πΌ β V) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
14 | 2, 3, 13 | syl2an 594 | . 2 β’ ((π β π β§ πΌ β π) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
15 | 1, 14 | eqtrid 2780 | 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 394 = wceq 1533 β wcel 2098 Vcvv 3473 {csn 4632 Γ cxp 5680 βcfv 6553 (class class class)co 7426 ringLModcrglmod 21064 βm cdsmm 21672 freeLMod cfrlm 21687 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-frlm 21688 |
This theorem is referenced by: frlmlmod 21690 frlmpws 21691 frlmlss 21692 frlmpwsfi 21693 frlmbas 21696 |
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