Theorem smatfval 31413

Description: Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)

Assertion

Ref	Expression
smatfval	⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))

Distinct variable groups:   𝑖,𝐾,𝑗   𝑖,𝐿,𝑗

Allowed substitution hints:   𝑀(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem smatfval

Dummy variables 𝑘 𝑙 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3416	. . . 4 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
2	1	3ad2ant3 1137	. . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → 𝑀 ∈ V)
3		coeq1 5711	. . . . 5 ⊢ (𝑚 = 𝑀 → (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩)) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩)))
4	3	mpoeq3dv 7268	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))) = (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
5		df-smat 31412	. . . 4 ⊢ subMat1 = (𝑚 ∈ V ↦ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑚 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
6		nnex 11801	. . . . 5 ⊢ ℕ ∈ V
7	6, 6	mpoex 7828	. . . 4 ⊢ (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))) ∈ V
8	4, 5, 7	fvmpt 6796	. . 3 ⊢ (𝑀 ∈ V → (subMat1‘𝑀) = (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
9	2, 8	syl 17	. 2 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (subMat1‘𝑀) = (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))))
10		breq2 5043	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑖 < 𝑘 ↔ 𝑖 < 𝐾))
11	10	ifbid 4448	. . . . . . 7 ⊢ (𝑘 = 𝐾 → if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)))
12	11	opeq1d 4776	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩ = ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩)
13	12	mpoeq3dv 7268	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))
14		breq2 5043	. . . . . . . 8 ⊢ (𝑙 = 𝐿 → (𝑗 < 𝑙 ↔ 𝑗 < 𝐿))
15	14	ifbid 4448	. . . . . . 7 ⊢ (𝑙 = 𝐿 → if(𝑗 < 𝑙, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)))
16	15	opeq2d 4777	. . . . . 6 ⊢ (𝑙 = 𝐿 → ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩ = ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
17	16	mpoeq3dv 7268	. . . . 5 ⊢ (𝑙 = 𝐿 → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
18	13, 17	sylan9eq 2791	. . . 4 ⊢ ((𝑘 = 𝐾 ∧ 𝑙 = 𝐿) → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
19	18	adantl 485	. . 3 ⊢ (((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩) = (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
20	19	coeq2d 5716	. 2 ⊢ (((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) ∧ (𝑘 = 𝐾 ∧ 𝑙 = 𝐿)) → (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩)) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
21		simp1 1138	. 2 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → 𝐾 ∈ ℕ)
22		simp2 1139	. 2 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → 𝐿 ∈ ℕ)
23		simp3 1140	. . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → 𝑀 ∈ 𝑉)
24	6, 6	mpoex 7828	. . . 4 ⊢ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∈ V
25	24	a1i 11	. . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∈ V)
26		coexg 7685	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∈ V) → (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ∈ V)
27	23, 25, 26	syl2anc 587	. 2 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ∈ V)
28	9, 20, 21, 22, 27	ovmpod 7339	1 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  Vcvv 3398  ifcif 4425  ⟨cop 4533   class class class wbr 5039   ∘ ccom 5540  ‘cfv 6358  (class class class)co 7191   ∈ cmpo 7193  1c1 10695   + caddc 10697   < clt 10832  ℕcn 11795  subMat1csmat 31411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-1cn 10752  ax-addcl 10754

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-nn 11796  df-smat 31412

This theorem is referenced by:  smatrcl  31414  smatlem  31415