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Theorem smatfval 33794
Description: Value of the submatrix. (Contributed by Thierry Arnoux, 19-Aug-2020.)
Assertion
Ref Expression
smatfval ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (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.
StepHypRef Expression
1 elex 3501 . . . 4 (𝑀𝑉𝑀 ∈ V)
213ad2ant3 1136 . . 3 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → 𝑀 ∈ V)
3 coeq1 5868 . . . . 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))⟩)))
43mpoeq3dv 7512 . . . 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 33793 . . . 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 12272 . . . . 5 ℕ ∈ V
76, 6mpoex 8104 . . . 4 (𝑘 ∈ ℕ, 𝑙 ∈ ℕ ↦ (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)), if(𝑗 < 𝑙, 𝑗, (𝑗 + 1))⟩))) ∈ V
84, 5, 7fvmpt 7016 . . 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))⟩))))
92, 8syl 17 . 2 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (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 5147 . . . . . . . 8 (𝑘 = 𝐾 → (𝑖 < 𝑘𝑖 < 𝐾))
1110ifbid 4549 . . . . . . 7 (𝑘 = 𝐾 → if(𝑖 < 𝑘, 𝑖, (𝑖 + 1)) = if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)))
1211opeq1d 4879 . . . . . 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))⟩)
1312mpoeq3dv 7512 . . . . 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 5147 . . . . . . . 8 (𝑙 = 𝐿 → (𝑗 < 𝑙𝑗 < 𝐿))
1514ifbid 4549 . . . . . . 7 (𝑙 = 𝐿 → if(𝑗 < 𝑙, 𝑗, (𝑗 + 1)) = if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)))
1615opeq2d 4880 . . . . . 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))⟩)
1716mpoeq3dv 7512 . . . . 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))⟩))
1813, 17sylan9eq 2797 . . . 4 ((𝑘 = 𝐾𝑙 = 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))⟩))
1918adantl 481 . . 3 (((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) ∧ (𝑘 = 𝐾𝑙 = 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))⟩))
2019coeq2d 5873 . 2 (((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) ∧ (𝑘 = 𝐾𝑙 = 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 1137 . 2 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → 𝐾 ∈ ℕ)
22 simp2 1138 . 2 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → 𝐿 ∈ ℕ)
23 simp3 1139 . . 3 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → 𝑀𝑉)
246, 6mpoex 8104 . . . 4 (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∈ V
2524a1i 11 . . 3 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∈ V)
26 coexg 7951 . . 3 ((𝑀𝑉 ∧ (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)
2723, 25, 26syl2anc 584 . 2 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) ∈ V)
289, 20, 21, 22, 27ovmpod 7585 1 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝑀𝑉) → (𝐾(subMat1‘𝑀)𝐿) = (𝑀 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  ifcif 4525  cop 4632   class class class wbr 5143  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  1c1 11156   + caddc 11158   < clt 11295  cn 12266  subMat1csmat 33792
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-1cn 11213  ax-addcl 11215
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-ral 3062  df-rex 3071  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-op 4633  df-uni 4908  df-iun 4993  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-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-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-smat 33793
This theorem is referenced by:  smatrcl  33795  smatlem  33796
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