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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 3462 | . . 3 β’ (π β π β π β V) | |
3 | elex 3462 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | id 22 | . . . . 5 β’ (π = π β π = π ) | |
5 | fveq2 6843 | . . . . . . 7 β’ (π = π β (ringLModβπ) = (ringLModβπ )) | |
6 | 5 | sneqd 4599 | . . . . . 6 β’ (π = π β {(ringLModβπ)} = {(ringLModβπ )}) |
7 | 6 | xpeq2d 5664 | . . . . 5 β’ (π = π β (π Γ {(ringLModβπ)}) = (π Γ {(ringLModβπ )})) |
8 | 4, 7 | oveq12d 7376 | . . . 4 β’ (π = π β (π βm (π Γ {(ringLModβπ)})) = (π βm (π Γ {(ringLModβπ )}))) |
9 | xpeq1 5648 | . . . . 5 β’ (π = πΌ β (π Γ {(ringLModβπ )}) = (πΌ Γ {(ringLModβπ )})) | |
10 | 9 | oveq2d 7374 | . . . 4 β’ (π = πΌ β (π βm (π Γ {(ringLModβπ )})) = (π βm (πΌ Γ {(ringLModβπ )}))) |
11 | df-frlm 21169 | . . . 4 β’ freeLMod = (π β V, π β V β¦ (π βm (π Γ {(ringLModβπ)}))) | |
12 | ovex 7391 | . . . 4 β’ (π βm (πΌ Γ {(ringLModβπ )})) β V | |
13 | 8, 10, 11, 12 | ovmpo 7516 | . . 3 β’ ((π β V β§ πΌ β V) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
14 | 2, 3, 13 | syl2an 597 | . 2 β’ ((π β π β§ πΌ β π) β (π freeLMod πΌ) = (π βm (πΌ Γ {(ringLModβπ )}))) |
15 | 1, 14 | eqtrid 2785 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 {csn 4587 Γ cxp 5632 βcfv 6497 (class class class)co 7358 ringLModcrglmod 20646 βm cdsmm 21153 freeLMod cfrlm 21168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-frlm 21169 |
This theorem is referenced by: frlmlmod 21171 frlmpws 21172 frlmlss 21173 frlmpwsfi 21174 frlmbas 21177 |
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