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Mirrors > Home > MPE Home > Th. List > pwspjmhm | Structured version Visualization version GIF version |
Description: A projection from a structure power of a monoid to the monoid itself is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
pwspjmhm.y | β’ π = (π βs πΌ) |
pwspjmhm.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
pwspjmhm | β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π₯ β π΅ β¦ (π₯βπ΄)) β (π MndHom π )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 β’ ((Scalarβπ )Xs(πΌ Γ {π })) = ((Scalarβπ )Xs(πΌ Γ {π })) | |
2 | eqid 2728 | . . 3 β’ (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) | |
3 | simp2 1134 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β πΌ β π) | |
4 | fvexd 6917 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (Scalarβπ ) β V) | |
5 | fconst6g 6791 | . . . 4 β’ (π β Mnd β (πΌ Γ {π }):πΌβΆMnd) | |
6 | 5 | 3ad2ant1 1130 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (πΌ Γ {π }):πΌβΆMnd) |
7 | simp3 1135 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β π΄ β πΌ) | |
8 | 1, 2, 3, 4, 6, 7 | prdspjmhm 18795 | . 2 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π₯ β (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) β¦ (π₯βπ΄)) β (((Scalarβπ )Xs(πΌ Γ {π })) MndHom ((πΌ Γ {π })βπ΄))) |
9 | pwspjmhm.b | . . . 4 β’ π΅ = (Baseβπ) | |
10 | pwspjmhm.y | . . . . . . 7 β’ π = (π βs πΌ) | |
11 | eqid 2728 | . . . . . . 7 β’ (Scalarβπ ) = (Scalarβπ ) | |
12 | 10, 11 | pwsval 17477 | . . . . . 6 β’ ((π β Mnd β§ πΌ β π) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
13 | 12 | 3adant3 1129 | . . . . 5 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β π = ((Scalarβπ )Xs(πΌ Γ {π }))) |
14 | 13 | fveq2d 6906 | . . . 4 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (Baseβπ) = (Baseβ((Scalarβπ )Xs(πΌ Γ {π })))) |
15 | 9, 14 | eqtrid 2780 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β π΅ = (Baseβ((Scalarβπ )Xs(πΌ Γ {π })))) |
16 | 15 | mpteq1d 5247 | . 2 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π₯ β π΅ β¦ (π₯βπ΄)) = (π₯ β (Baseβ((Scalarβπ )Xs(πΌ Γ {π }))) β¦ (π₯βπ΄))) |
17 | fvconst2g 7220 | . . . . 5 β’ ((π β Mnd β§ π΄ β πΌ) β ((πΌ Γ {π })βπ΄) = π ) | |
18 | 17 | 3adant2 1128 | . . . 4 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β ((πΌ Γ {π })βπ΄) = π ) |
19 | 18 | eqcomd 2734 | . . 3 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β π = ((πΌ Γ {π })βπ΄)) |
20 | 13, 19 | oveq12d 7444 | . 2 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π MndHom π ) = (((Scalarβπ )Xs(πΌ Γ {π })) MndHom ((πΌ Γ {π })βπ΄))) |
21 | 8, 16, 20 | 3eltr4d 2844 | 1 β’ ((π β Mnd β§ πΌ β π β§ π΄ β πΌ) β (π₯ β π΅ β¦ (π₯βπ΄)) β (π MndHom π )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3473 {csn 4632 β¦ cmpt 5235 Γ cxp 5680 βΆwf 6549 βcfv 6553 (class class class)co 7426 Basecbs 17189 Scalarcsca 17245 Xscprds 17436 βs cpws 17437 Mndcmnd 18703 MndHom cmhm 18747 |
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-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-map 8855 df-ixp 8925 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-struct 17125 df-slot 17160 df-ndx 17172 df-base 17190 df-plusg 17255 df-mulr 17256 df-sca 17258 df-vsca 17259 df-ip 17260 df-tset 17261 df-ple 17262 df-ds 17264 df-hom 17266 df-cco 17267 df-0g 17432 df-prds 17438 df-pws 17440 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-mhm 18749 |
This theorem is referenced by: pwsmulg 19088 |
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