smadiadetlem1a - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > smadiadetlem1a		Structured version Visualization version GIF version

Theorem smadiadetlem1a 21812

Description: Lemma 1a for smadiadet 21819: The summands 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.)

**Hypotheses**
Ref	Expression
marep01ma.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marep01ma.b	⊢ 𝐵 = (Base‘𝐴)
marep01ma.r	⊢ 𝑅 ∈ CRing
marep01ma.0	⊢ 0 = (0_g‘𝑅)
marep01ma.1	⊢ 1 = (1_r‘𝑅)
smadiadetlem.p	⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
smadiadetlem.g	⊢ 𝐺 = (mulGrp‘𝑅)
madetminlem.y	⊢ 𝑌 = (ℤRHom‘𝑅)
madetminlem.s	⊢ 𝑆 = (pmSgn‘𝑁)
madetminlem.t	⊢ · = (._r‘𝑅)

**Assertion**
Ref	Expression
smadiadetlem1a	⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )

Distinct variable groups: 𝑖,𝑗,𝑛,𝐵 𝑖,𝑞,𝐾,𝑗,𝑛 𝑖,𝐿,𝑗,𝑛,𝑞 𝑖,𝑀,𝑗,𝑛 𝑖,𝑁,𝑗,𝑛 𝑃,𝑖,𝑗,𝑛,𝑞 𝑅,𝑖,𝑗,𝑛 1 ,𝑖,𝑗,𝑛 0 ,𝑖,𝑗,𝑛 𝑛,𝐺 𝑛,𝑝,𝐵 𝐾,𝑝 𝐿,𝑝 𝑀,𝑝 𝑁,𝑝 𝑃,𝑝 𝑅,𝑝,𝑖,𝑗 𝑞,𝑝 Allowed substitution hints: 𝐴(𝑖,𝑗,𝑛,𝑞,𝑝) 𝐵(𝑞) 𝑅(𝑞) 𝑆(𝑖,𝑗,𝑛,𝑞,𝑝) · (𝑖,𝑗,𝑛,𝑞,𝑝) 1 (𝑞,𝑝) 𝐺(𝑖,𝑗,𝑞,𝑝) 𝑀(𝑞) 𝑁(𝑞) 𝑌(𝑖,𝑗,𝑛,𝑞,𝑝) 0 (𝑞,𝑝)

**Proof of Theorem smadiadetlem1a**
Step	Hyp	Ref	Expression
1		marep01ma.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marep01ma.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
3		marep01ma.r	. . . . . . 7 ⊢ 𝑅 ∈ CRing
4		marep01ma.0	. . . . . . 7 ⊢ 0 = (0_g‘𝑅)
5		marep01ma.1	. . . . . . 7 ⊢ 1 = (1_r‘𝑅)
6		smadiadetlem.p	. . . . . . 7 ⊢ 𝑃 = (Base‘(SymGrp‘𝑁))
7		smadiadetlem.g	. . . . . . 7 ⊢ 𝐺 = (mulGrp‘𝑅)
8	1, 2, 3, 4, 5, 6, 7	smadiadetlem0 21810	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))) = 0 ))
9	8	imp 407	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))) = 0 )
10	9	oveq2d 7291	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))) = (((𝑌 ∘ 𝑆)‘𝑝) · 0 ))
11	10	mpteq2dva 5174	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛)))))) = (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · 0 )))
12	11	oveq2d 7291	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · 0 ))))
13		crngring 19795	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	3, 13	mp1i 13	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑅 ∈ Ring)
15	1, 2	matrcl 21559	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
16	15	simpld 495	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
17	16	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → 𝑁 ∈ Fin)
18	17	adantr 481	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑁 ∈ Fin)
19		eldifi 4061	. . . . . . 7 ⊢ (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) → 𝑝 ∈ 𝑃)
20	19	adantl 482	. . . . . 6 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → 𝑝 ∈ 𝑃)
21		madetminlem.s	. . . . . . 7 ⊢ 𝑆 = (pmSgn‘𝑁)
22		madetminlem.y	. . . . . . 7 ⊢ 𝑌 = (ℤRHom‘𝑅)
23	6, 21, 22	zrhcopsgnelbas 20800	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin ∧ 𝑝 ∈ 𝑃) → ((𝑌 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅))
24	14, 18, 20, 23	syl3anc 1370	. . . . 5 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → ((𝑌 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅))
25		eqid 2738	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
26		madetminlem.t	. . . . . 6 ⊢ · = (._r‘𝑅)
27	25, 26, 4	ringrz 19827	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ ((𝑌 ∘ 𝑆)‘𝑝) ∈ (Base‘𝑅)) → (((𝑌 ∘ 𝑆)‘𝑝) · 0 ) = 0 )
28	14, 24, 27	syl2anc 584	. . . 4 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) ∧ 𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿})) → (((𝑌 ∘ 𝑆)‘𝑝) · 0 ) = 0 )
29	28	mpteq2dva 5174	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · 0 )) = (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ 0 ))
30	29	oveq2d 7291	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · 0 ))) = (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ 0 )))
31		ringmnd 19793	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
32	3, 13, 31	mp2b 10	. . 3 ⊢ 𝑅 ∈ Mnd
33	6	fvexi 6788	. . . 4 ⊢ 𝑃 ∈ V
34		difexg 5251	. . . 4 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ∈ V)
35	33, 34	mp1i 13	. . 3 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ∈ V)
36	4	gsumz 18474	. . 3 ⊢ ((𝑅 ∈ Mnd ∧ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ∈ V) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ 0 )) = 0 )
37	32, 35, 36	sylancr 587	. 2 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ 0 )) = 0 )
38	12, 30, 37	3eqtrd 2782	1 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁) → (𝑅 Σ_g (𝑝 ∈ (𝑃 ∖ {𝑞 ∈ 𝑃 ∣ (𝑞‘𝐾) = 𝐿}) ↦ (((𝑌 ∘ 𝑆)‘𝑝) · (𝐺 Σ_g (𝑛 ∈ 𝑁 ↦ (𝑛(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 1 , 0 ), (𝑖𝑀𝑗)))(𝑝‘𝑛))))))) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {crab 3068 Vcvv 3432 ∖ cdif 3884 ifcif 4459 ↦ cmpt 5157 ∘ ccom 5593 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 Basecbs 16912 ._rcmulr 16963 0_gc0g 17150 Σ_g cgsu 17151 Mndcmnd 18385 SymGrpcsymg 18974 pmSgncpsgn 19097 mulGrpcmgp 19720 1_rcur 19737 Ringcrg 19783 CRingccrg 19784 ℤRHomczrh 20701 Mat cmat 21554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-xor 1507 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-word 14218 df-lsw 14266 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-reverse 14472 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-0g 17152 df-gsum 17153 df-prds 17158 df-pws 17160 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-submnd 18431 df-efmnd 18508 df-grp 18580 df-minusg 18581 df-mulg 18701 df-subg 18752 df-ghm 18832 df-gim 18875 df-cntz 18923 df-oppg 18950 df-symg 18975 df-pmtr 19050 df-psgn 19099 df-cmn 19388 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-rnghom 19959 df-subrg 20022 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-dsmm 20939 df-frlm 20954 df-mat 21555

This theorem is referenced by: smadiadetlem2 21813

W3C validator