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Mirrors > Home > MPE Home > Th. List > ascldimul | Structured version Visualization version GIF version |
Description: The algebra scalars function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.) |
Ref | Expression |
---|---|
ascldimul.a | ⊢ 𝐴 = (algSc‘𝑊) |
ascldimul.f | ⊢ 𝐹 = (Scalar‘𝑊) |
ascldimul.k | ⊢ 𝐾 = (Base‘𝐹) |
ascldimul.t | ⊢ × = (.r‘𝑊) |
ascldimul.s | ⊢ · = (.r‘𝐹) |
Ref | Expression |
---|---|
ascldimul | ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴‘𝑅) × (𝐴‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | assalmod 20087 | . . . 4 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
2 | 1 | 3ad2ant1 1128 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → 𝑊 ∈ LMod) |
3 | simp2 1132 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → 𝑅 ∈ 𝐾) | |
4 | simp3 1133 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → 𝑆 ∈ 𝐾) | |
5 | assaring 20088 | . . . . 5 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) | |
6 | 5 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → 𝑊 ∈ Ring) |
7 | eqid 2820 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
8 | eqid 2820 | . . . . 5 ⊢ (1r‘𝑊) = (1r‘𝑊) | |
9 | 7, 8 | ringidcl 19313 | . . . 4 ⊢ (𝑊 ∈ Ring → (1r‘𝑊) ∈ (Base‘𝑊)) |
10 | 6, 9 | syl 17 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (1r‘𝑊) ∈ (Base‘𝑊)) |
11 | ascldimul.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
12 | eqid 2820 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
13 | ascldimul.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
14 | ascldimul.s | . . . 4 ⊢ · = (.r‘𝐹) | |
15 | 7, 11, 12, 13, 14 | lmodvsass 19654 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾 ∧ (1r‘𝑊) ∈ (Base‘𝑊))) → ((𝑅 · 𝑆)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑅( ·𝑠 ‘𝑊)(𝑆( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
16 | 2, 3, 4, 10, 15 | syl13anc 1367 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → ((𝑅 · 𝑆)( ·𝑠 ‘𝑊)(1r‘𝑊)) = (𝑅( ·𝑠 ‘𝑊)(𝑆( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
17 | 11 | lmodring 19637 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
18 | 1, 17 | syl 17 | . . . 4 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring) |
19 | 13, 14 | ringcl 19306 | . . . 4 ⊢ ((𝐹 ∈ Ring ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝑅 · 𝑆) ∈ 𝐾) |
20 | 18, 19 | syl3an1 1158 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝑅 · 𝑆) ∈ 𝐾) |
21 | ascldimul.a | . . . 4 ⊢ 𝐴 = (algSc‘𝑊) | |
22 | 21, 11, 13, 12, 8 | asclval 20104 | . . 3 ⊢ ((𝑅 · 𝑆) ∈ 𝐾 → (𝐴‘(𝑅 · 𝑆)) = ((𝑅 · 𝑆)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
23 | 20, 22 | syl 17 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝑅 · 𝑆)( ·𝑠 ‘𝑊)(1r‘𝑊))) |
24 | 21, 11, 5, 1, 13, 7 | asclf 20106 | . . . . . 6 ⊢ (𝑊 ∈ AssAlg → 𝐴:𝐾⟶(Base‘𝑊)) |
25 | 24 | ffvelrnda 6844 | . . . . 5 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑆 ∈ 𝐾) → (𝐴‘𝑆) ∈ (Base‘𝑊)) |
26 | 25 | 3adant2 1126 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘𝑆) ∈ (Base‘𝑊)) |
27 | ascldimul.t | . . . . 5 ⊢ × = (.r‘𝑊) | |
28 | 21, 11, 13, 7, 27, 12 | asclmul1 20109 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ (𝐴‘𝑆) ∈ (Base‘𝑊)) → ((𝐴‘𝑅) × (𝐴‘𝑆)) = (𝑅( ·𝑠 ‘𝑊)(𝐴‘𝑆))) |
29 | 26, 28 | syld3an3 1404 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → ((𝐴‘𝑅) × (𝐴‘𝑆)) = (𝑅( ·𝑠 ‘𝑊)(𝐴‘𝑆))) |
30 | 21, 11, 13, 12, 8 | asclval 20104 | . . . . 5 ⊢ (𝑆 ∈ 𝐾 → (𝐴‘𝑆) = (𝑆( ·𝑠 ‘𝑊)(1r‘𝑊))) |
31 | 30 | 3ad2ant3 1130 | . . . 4 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘𝑆) = (𝑆( ·𝑠 ‘𝑊)(1r‘𝑊))) |
32 | 31 | oveq2d 7165 | . . 3 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝑅( ·𝑠 ‘𝑊)(𝐴‘𝑆)) = (𝑅( ·𝑠 ‘𝑊)(𝑆( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
33 | 29, 32 | eqtrd 2855 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → ((𝐴‘𝑅) × (𝐴‘𝑆)) = (𝑅( ·𝑠 ‘𝑊)(𝑆( ·𝑠 ‘𝑊)(1r‘𝑊)))) |
34 | 16, 23, 33 | 3eqtr4d 2865 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ 𝑅 ∈ 𝐾 ∧ 𝑆 ∈ 𝐾) → (𝐴‘(𝑅 · 𝑆)) = ((𝐴‘𝑅) × (𝐴‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 .rcmulr 16561 Scalarcsca 16563 ·𝑠 cvsca 16564 1rcur 19246 Ringcrg 19292 LModclmod 19629 AssAlgcasa 20077 algSccascl 20079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-plusg 16573 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-mgp 19235 df-ur 19247 df-ring 19294 df-lmod 19631 df-assa 20080 df-ascl 20082 |
This theorem is referenced by: asclrhm 20114 |
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