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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmod1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for lmod1 47262. (Contributed by AV, 28-Apr-2019.) |
Ref | Expression |
---|---|
lmod1.m | β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) |
Ref | Expression |
---|---|
lmod1lem1 | β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ)πΌ) β {πΌ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6905 | . . . . . 6 β’ (Baseβπ ) β V | |
2 | snex 5432 | . . . . . . 7 β’ {πΌ} β V | |
3 | 2 | a1i 11 | . . . . . 6 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β {πΌ} β V) |
4 | mpoexga 8067 | . . . . . 6 β’ (((Baseβπ ) β V β§ {πΌ} β V) β (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦) β V) | |
5 | 1, 3, 4 | sylancr 586 | . . . . 5 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦) β V) |
6 | lmod1.m | . . . . . 6 β’ π = ({β¨(Baseβndx), {πΌ}β©, β¨(+gβndx), {β¨β¨πΌ, πΌβ©, πΌβ©}β©, β¨(Scalarβndx), π β©} βͺ {β¨( Β·π βndx), (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)β©}) | |
7 | 6 | lmodvsca 17279 | . . . . 5 β’ ((π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦) β V β (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦) = ( Β·π βπ)) |
8 | 5, 7 | syl 17 | . . . 4 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦) = ( Β·π βπ)) |
9 | 8 | eqcomd 2737 | . . 3 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β ( Β·π βπ) = (π₯ β (Baseβπ ), π¦ β {πΌ} β¦ π¦)) |
10 | simprr 770 | . . 3 β’ (((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β§ (π₯ = π β§ π¦ = πΌ)) β π¦ = πΌ) | |
11 | simp3 1137 | . . 3 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β π β (Baseβπ )) | |
12 | snidg 4663 | . . . 4 β’ (πΌ β π β πΌ β {πΌ}) | |
13 | 12 | 3ad2ant1 1132 | . . 3 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β πΌ β {πΌ}) |
14 | 9, 10, 11, 13, 13 | ovmpod 7563 | . 2 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ)πΌ) = πΌ) |
15 | 14, 13 | eqeltrd 2832 | 1 β’ ((πΌ β π β§ π β Ring β§ π β (Baseβπ )) β (π( Β·π βπ)πΌ) β {πΌ}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 Vcvv 3473 βͺ cun 3947 {csn 4629 {ctp 4633 β¨cop 4635 βcfv 6544 (class class class)co 7412 β cmpo 7414 ndxcnx 17131 Basecbs 17149 +gcplusg 17202 Scalarcsca 17205 Β·π cvsca 17206 Ringcrg 20128 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-sca 17218 df-vsca 17219 |
This theorem is referenced by: lmod1 47262 |
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