Theorem dmatelnd 22486

Description: An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)

Hypotheses

Ref	Expression
dmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
dmatid.b	⊢ 𝐵 = (Base‘𝐴)
dmatid.0	⊢ 0 = (0g‘𝑅)
dmatid.d	⊢ 𝐷 = (𝑁 DMat 𝑅)

Assertion

Ref	Expression
dmatelnd	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 )

Proof of Theorem dmatelnd

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatid.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		dmatid.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
4		dmatid.d	. . . . 5 ⊢ 𝐷 = (𝑁 DMat 𝑅)
5	1, 2, 3, 4	dmatel 22483	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ))))
6		neeq1 2997	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖 ≠ 𝑗 ↔ 𝐼 ≠ 𝑗))
7		oveq1 7370	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
8	7	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
9	6, 8	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 )))
10		neeq2 2998	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼 ≠ 𝑗 ↔ 𝐼 ≠ 𝐽))
11		oveq2 7371	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
12	11	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
13	10, 12	imbi12d 345	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
14	9, 13	rspc2v 3578	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
15	14	com23 86	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼 ≠ 𝐽 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16	15	3impia 1123	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 ))
17	16	com12 32	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))
18	17	2a1i 12	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐵 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))))
19	18	impd 411	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 )))
20	5, 19	sylbid 241	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 )))
21	20	3impia 1123	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))
22	21	imp 407	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 )

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Fincfn 8890  Basecbs 17177  0_gc0g 17400  Ringcrg 20212   Mat cmat 22397   DMat cdmat 22478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-dmat 22480

This theorem is referenced by:  dmatmul  22487  dmatsubcl  22488