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Theorem matmulcell 22420

Description: Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.)

**Hypotheses**
Ref	Expression
matmulcell.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
matmulcell.b	⊢ 𝐵 = (Base‘𝐴)
matmulcell.m	⊢ × = (._r‘𝐴)

**Assertion**
Ref	Expression
matmulcell	⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(._r‘𝑅)(𝑗𝑌𝐽)))))

Distinct variable groups: 𝐵,𝑗 𝑗,𝐼 𝑗,𝐽 𝑗,𝑁 𝑅,𝑗 𝑗,𝑋 𝑗,𝑌 Allowed substitution hints: 𝐴(𝑗) × (𝑗)

**Proof of Theorem matmulcell**
Step	Hyp	Ref	Expression
1		matmulcell.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		matmulcell.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 22387	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4		matmulcell.m	. . . . . . . . . 10 ⊢ × = (._r‘𝐴)
5		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
6	1, 5	matmulr 22413	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
7	4, 6	eqtr4id 2791	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
8	7	a1d 25	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑅 ∈ Ring → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)))
9	3, 8	syl 17	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (𝑅 ∈ Ring → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)))
10	9	adantr 480	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑅 ∈ Ring → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)))
11	10	impcom 407	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
12	11	3adant3 1133	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩))
13	12	oveqd 7377	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝑋 × 𝑌) = (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌))
14	13	oveqd 7377	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝐼(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌)𝐽))
15		eqid 2737	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2737	. . 3 ⊢ (._r‘𝑅) = (._r‘𝑅)
17		simp1 1137	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring)
18	3	simpld 494	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
19	18	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑁 ∈ Fin)
20	19	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin)
21	1, 15, 2	matbas2i 22397	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
22	21	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
23	22	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑋 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
24	1, 15, 2	matbas2i 22397	. . . . 5 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
25	24	adantl 481	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
26	25	3ad2ant2 1135	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
27		simp3l 1203	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐼 ∈ 𝑁)
28		simp3r 1204	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝐽 ∈ 𝑁)
29	5, 15, 16, 17, 20, 20, 20, 23, 26, 27, 28	mamufv 22369	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝑌)𝐽) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(._r‘𝑅)(𝑗𝑌𝐽)))))
30	14, 29	eqtrd 2772	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝐼𝑋𝑗)(._r‘𝑅)(𝑗𝑌𝐽)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⟨cotp 4576 ↦ cmpt 5167 × cxp 5622 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 Fincfn 8886 Basecbs 17170 ._rcmulr 17212 Σ_g cgsu 17394 Ringcrg 20205 maMul cmmul 22365 Mat cmat 22382

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-hom 17235 df-cco 17236 df-0g 17395 df-prds 17401 df-pws 17403 df-sra 21160 df-rgmod 21161 df-dsmm 21722 df-frlm 21737 df-mamu 22366 df-mat 22383

This theorem is referenced by: mat1mhm 22459 scmatscm 22488 cpmatmcl 22694

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