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Mirrors > Home > MPE Home > Th. List > pwsdiaglmhm | Structured version Visualization version GIF version |
Description: Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
Ref | Expression |
---|---|
pwsdiaglmhm.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsdiaglmhm.b | ⊢ 𝐵 = (Base‘𝑅) |
pwsdiaglmhm.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) |
Ref | Expression |
---|---|
pwsdiaglmhm | ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsdiaglmhm.b | . 2 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2728 | . 2 ⊢ ( ·𝑠 ‘𝑅) = ( ·𝑠 ‘𝑅) | |
3 | eqid 2728 | . 2 ⊢ ( ·𝑠 ‘𝑌) = ( ·𝑠 ‘𝑌) | |
4 | eqid 2728 | . 2 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
5 | eqid 2728 | . 2 ⊢ (Scalar‘𝑌) = (Scalar‘𝑌) | |
6 | eqid 2728 | . 2 ⊢ (Base‘(Scalar‘𝑅)) = (Base‘(Scalar‘𝑅)) | |
7 | simpl 482 | . 2 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ LMod) | |
8 | pwsdiaglmhm.y | . . 3 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
9 | 8 | pwslmod 20854 | . 2 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝑌 ∈ LMod) |
10 | 8, 4 | pwssca 17478 | . . 3 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → (Scalar‘𝑅) = (Scalar‘𝑌)) |
11 | 10 | eqcomd 2734 | . 2 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → (Scalar‘𝑌) = (Scalar‘𝑅)) |
12 | lmodgrp 20750 | . . 3 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp) | |
13 | pwsdiaglmhm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) | |
14 | 8, 1, 13 | pwsdiagghm 19198 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
15 | 12, 14 | sylan 579 | . 2 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 GrpHom 𝑌)) |
16 | simplr 768 | . . . 4 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → 𝐼 ∈ 𝑊) | |
17 | 1, 4, 2, 6 | lmodvscl 20761 | . . . . . 6 ⊢ ((𝑅 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵) → (𝑎( ·𝑠 ‘𝑅)𝑏) ∈ 𝐵) |
18 | 17 | 3expb 1118 | . . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝑎( ·𝑠 ‘𝑅)𝑏) ∈ 𝐵) |
19 | 18 | adantlr 714 | . . . 4 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝑎( ·𝑠 ‘𝑅)𝑏) ∈ 𝐵) |
20 | 13 | fvdiagfn 8910 | . . . 4 ⊢ ((𝐼 ∈ 𝑊 ∧ (𝑎( ·𝑠 ‘𝑅)𝑏) ∈ 𝐵) → (𝐹‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝐼 × {(𝑎( ·𝑠 ‘𝑅)𝑏)})) |
21 | 16, 19, 20 | syl2anc 583 | . . 3 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝐼 × {(𝑎( ·𝑠 ‘𝑅)𝑏)})) |
22 | 13 | fvdiagfn 8910 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑏) = (𝐼 × {𝑏})) |
23 | 22 | ad2ant2l 745 | . . . . 5 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝐹‘𝑏) = (𝐼 × {𝑏})) |
24 | 23 | oveq2d 7436 | . . . 4 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝑎( ·𝑠 ‘𝑌)(𝐹‘𝑏)) = (𝑎( ·𝑠 ‘𝑌)(𝐼 × {𝑏}))) |
25 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
26 | simpll 766 | . . . . 5 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → 𝑅 ∈ LMod) | |
27 | simprl 770 | . . . . 5 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ (Base‘(Scalar‘𝑅))) | |
28 | 8, 1, 25 | pwsdiagel 17479 | . . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ 𝑏 ∈ 𝐵) → (𝐼 × {𝑏}) ∈ (Base‘𝑌)) |
29 | 28 | adantrl 715 | . . . . 5 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝐼 × {𝑏}) ∈ (Base‘𝑌)) |
30 | 8, 25, 2, 3, 4, 6, 26, 16, 27, 29 | pwsvscafval 17476 | . . . 4 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝑎( ·𝑠 ‘𝑌)(𝐼 × {𝑏})) = ((𝐼 × {𝑎}) ∘f ( ·𝑠 ‘𝑅)(𝐼 × {𝑏}))) |
31 | id 22 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ 𝑊) | |
32 | vex 3475 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝑎 ∈ V) |
34 | vex 3475 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
35 | 34 | a1i 11 | . . . . . 6 ⊢ (𝐼 ∈ 𝑊 → 𝑏 ∈ V) |
36 | 31, 33, 35 | ofc12 7713 | . . . . 5 ⊢ (𝐼 ∈ 𝑊 → ((𝐼 × {𝑎}) ∘f ( ·𝑠 ‘𝑅)(𝐼 × {𝑏})) = (𝐼 × {(𝑎( ·𝑠 ‘𝑅)𝑏)})) |
37 | 36 | ad2antlr 726 | . . . 4 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → ((𝐼 × {𝑎}) ∘f ( ·𝑠 ‘𝑅)(𝐼 × {𝑏})) = (𝐼 × {(𝑎( ·𝑠 ‘𝑅)𝑏)})) |
38 | 24, 30, 37 | 3eqtrd 2772 | . . 3 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝑎( ·𝑠 ‘𝑌)(𝐹‘𝑏)) = (𝐼 × {(𝑎( ·𝑠 ‘𝑅)𝑏)})) |
39 | 21, 38 | eqtr4d 2771 | . 2 ⊢ (((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑅)) ∧ 𝑏 ∈ 𝐵)) → (𝐹‘(𝑎( ·𝑠 ‘𝑅)𝑏)) = (𝑎( ·𝑠 ‘𝑌)(𝐹‘𝑏))) |
40 | 1, 2, 3, 4, 5, 6, 7, 9, 11, 15, 39 | islmhmd 20924 | 1 ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {csn 4629 ↦ cmpt 5231 × cxp 5676 ‘cfv 6548 (class class class)co 7420 ∘f cof 7683 Basecbs 17180 Scalarcsca 17236 ·𝑠 cvsca 17237 ↑s cpws 17428 Grpcgrp 18890 GrpHom cghm 19167 LModclmod 20743 LMHom clmhm 20904 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 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 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-hom 17257 df-cco 17258 df-0g 17423 df-prds 17429 df-pws 17431 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-grp 18893 df-minusg 18894 df-ghm 19168 df-mgp 20075 df-ur 20122 df-ring 20175 df-lmod 20745 df-lmhm 20907 |
This theorem is referenced by: pwslnmlem1 42516 |
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