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| Mirrors > Home > MPE Home > Th. List > smadiadetlem0 | Structured version Visualization version GIF version | ||
| Description: Lemma 0 for smadiadet 22676: The products of the Leibniz' formula vanish for all permutations fixing the index of the row containing the 0's and the 1 to the column with the 1. (Contributed by AV, 3-Jan-2019.) |
| Ref | Expression |
|---|---|
| marep01ma.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
| marep01ma.b | ⊢ 𝐵 = (Base‘𝐴) |
| marep01ma.r | ⊢ 𝑅 ∈ CRing |
| marep01ma.0 | ⊢ 0 = (0g‘𝑅) |
| marep01ma.1 | ⊢ 1 = (1r‘𝑅) |
| smadiadetlem.p | ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) |
| smadiadetlem.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
| Ref | Expression |
|---|---|
| smadiadetlem0 | ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)))) = 0 )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | smadiadetlem.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 2 | marep01ma.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 3 | marep01ma.r | . . . 4 ⊢ 𝑅 ∈ CRing | |
| 4 | 3 | a1i 11 | . . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑅 ∈ CRing) |
| 5 | marep01ma.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
| 6 | marep01ma.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐴) | |
| 7 | 5, 6 | matrcl 22416 | . . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
| 8 | 7 | simpld 494 | . . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin) |
| 9 | 8 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → 𝑁 ∈ Fin) |
| 10 | 9 | adantr 480 | . . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑁 ∈ Fin) |
| 11 | crngring 20242 | . . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring) | |
| 12 | 3, 11 | mp1i 13 | . . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑅 ∈ Ring) |
| 13 | eldifi 4131 | . . . . . . 7 ⊢ (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → 𝑄 ∈ 𝑃) | |
| 14 | 13 | adantl 481 | . . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑄 ∈ 𝑃) |
| 15 | marep01ma.1 | . . . . . . . . 9 ⊢ 1 = (1r‘𝑅) | |
| 16 | 5, 6, 3, 2, 15 | marep01ma 22666 | . . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) |
| 17 | 16 | 3ad2ant1 1134 | . . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) |
| 18 | 17 | adantr 480 | . . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) |
| 19 | smadiadetlem.p | . . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁)) | |
| 20 | 5, 6, 19 | matepm2cl 22469 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑄 ∈ 𝑃 ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗))) ∈ 𝐵) → ∀𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) ∈ (Base‘𝑅)) |
| 21 | 12, 14, 18, 20 | syl3anc 1373 | . . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ∀𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) ∈ (Base‘𝑅)) |
| 22 | id 22 | . . . . . . . 8 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛) | |
| 23 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑄‘𝑚) = (𝑄‘𝑛)) | |
| 24 | 22, 23 | oveq12d 7449 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) = (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛))) |
| 25 | 24 | eleq1d 2826 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) ∈ (Base‘𝑅) ↔ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) ∈ (Base‘𝑅))) |
| 26 | 25 | rspccv 3619 | . . . . 5 ⊢ (∀𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) ∈ (Base‘𝑅) → (𝑛 ∈ 𝑁 → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) ∈ (Base‘𝑅))) |
| 27 | 21, 26 | syl 17 | . . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (𝑛 ∈ 𝑁 → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) ∈ (Base‘𝑅))) |
| 28 | 27 | imp 406 | . . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) ∧ 𝑛 ∈ 𝑁) → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) ∈ (Base‘𝑅)) |
| 29 | id 22 | . . . . 5 ⊢ (𝑛 = 𝑚 → 𝑛 = 𝑚) | |
| 30 | fveq2 6906 | . . . . 5 ⊢ (𝑛 = 𝑚 → (𝑄‘𝑛) = (𝑄‘𝑚)) | |
| 31 | 29, 30 | oveq12d 7449 | . . . 4 ⊢ (𝑛 = 𝑚 → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) = (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚))) |
| 32 | 31 | adantl 481 | . . 3 ⊢ ((((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) ∧ 𝑛 = 𝑚) → (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)) = (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚))) |
| 33 | 19, 2, 15 | symgmatr01 22660 | . . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) = 0 )) |
| 34 | 33 | 3adant1 1131 | . . . 4 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → ∃𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) = 0 )) |
| 35 | 34 | imp 406 | . . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ∃𝑚 ∈ 𝑁 (𝑚(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑚)) = 0 ) |
| 36 | 1, 2, 4, 10, 28, 32, 35 | gsummgp0 20315 | . 2 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)))) = 0 ) |
| 37 | 36 | ex 412 | 1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑄 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → (𝐺 Σg (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑄‘𝑛)))) = 0 )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∖ cdif 3948 ifcif 4525 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 Fincfn 8985 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 SymGrpcsymg 19386 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 CRingccrg 20231 Mat cmat 22411 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cntz 19335 df-symg 19387 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-sra 21172 df-rgmod 21173 df-dsmm 21752 df-frlm 21767 df-mat 22412 |
| This theorem is referenced by: smadiadetlem1a 22669 |
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