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Theorem smatlem 30203
Description: Lemma for the next theorems. (Contributed by Thierry Arnoux, 19-Aug-2020.)
Hypotheses
Ref Expression
smat.s 𝑆 = (𝐾(subMat1‘𝐴)𝐿)
smat.m (𝜑𝑀 ∈ ℕ)
smat.n (𝜑𝑁 ∈ ℕ)
smat.k (𝜑𝐾 ∈ (1...𝑀))
smat.l (𝜑𝐿 ∈ (1...𝑁))
smat.a (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))
smatlem.i (𝜑𝐼 ∈ ℕ)
smatlem.j (𝜑𝐽 ∈ ℕ)
smatlem.1 (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)
smatlem.2 (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)
Assertion
Ref Expression
smatlem (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Proof of Theorem smatlem
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smat.s . . . . . 6 𝑆 = (𝐾(subMat1‘𝐴)𝐿)
2 fz1ssnn 12579 . . . . . . . 8 (1...𝑀) ⊆ ℕ
3 smat.k . . . . . . . 8 (𝜑𝐾 ∈ (1...𝑀))
42, 3sseldi 3750 . . . . . . 7 (𝜑𝐾 ∈ ℕ)
5 fz1ssnn 12579 . . . . . . . 8 (1...𝑁) ⊆ ℕ
6 smat.l . . . . . . . 8 (𝜑𝐿 ∈ (1...𝑁))
75, 6sseldi 3750 . . . . . . 7 (𝜑𝐿 ∈ ℕ)
8 smat.a . . . . . . 7 (𝜑𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁))))
9 smatfval 30201 . . . . . . 7 ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝐴 ∈ (𝐵𝑚 ((1...𝑀) × (1...𝑁)))) → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
104, 7, 8, 9syl3anc 1476 . . . . . 6 (𝜑 → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
111, 10syl5eq 2817 . . . . 5 (𝜑𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
1211oveqd 6810 . . . 4 (𝜑 → (𝐼𝑆𝐽) = (𝐼(𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))𝐽))
13 df-ov 6796 . . . 4 (𝐼(𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))𝐽) = ((𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))‘⟨𝐼, 𝐽⟩)
1412, 13syl6eq 2821 . . 3 (𝜑 → (𝐼𝑆𝐽) = ((𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))‘⟨𝐼, 𝐽⟩))
15 smatlem.i . . . . . . 7 (𝜑𝐼 ∈ ℕ)
16 smatlem.j . . . . . . 7 (𝜑𝐽 ∈ ℕ)
1715, 16jca 501 . . . . . 6 (𝜑 → (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
18 opelxp 5286 . . . . . 6 (⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ) ↔ (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
1917, 18sylibr 224 . . . . 5 (𝜑 → ⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ))
20 eqid 2771 . . . . . 6 (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))⟩)
21 opex 5060 . . . . . 6 ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
2220, 21dmmpt2 7390 . . . . 5 dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (ℕ × ℕ)
2319, 22syl6eleqr 2861 . . . 4 (𝜑 → ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
2420mpt2fun 6909 . . . . 5 Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
25 fvco 6416 . . . . 5 ((Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) ∧ ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)) → ((𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))‘⟨𝐼, 𝐽⟩) = (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)))
2624, 25mpan 670 . . . 4 (⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) → ((𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))‘⟨𝐼, 𝐽⟩) = (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)))
2723, 26syl 17 . . 3 (𝜑 → ((𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))‘⟨𝐼, 𝐽⟩) = (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)))
2814, 27eqtrd 2805 . 2 (𝜑 → (𝐼𝑆𝐽) = (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)))
29 df-ov 6796 . . . . 5 (𝐼(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)𝐽) = ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)
30 breq1 4789 . . . . . . . . . 10 (𝑖 = 𝐼 → (𝑖 < 𝐾𝐼 < 𝐾))
31 id 22 . . . . . . . . . 10 (𝑖 = 𝐼𝑖 = 𝐼)
32 oveq1 6800 . . . . . . . . . 10 (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
3330, 31, 32ifbieq12d 4252 . . . . . . . . 9 (𝑖 = 𝐼 → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)))
3433opeq1d 4545 . . . . . . . 8 (𝑖 = 𝐼 → ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
35 breq1 4789 . . . . . . . . . 10 (𝑗 = 𝐽 → (𝑗 < 𝐿𝐽 < 𝐿))
36 id 22 . . . . . . . . . 10 (𝑗 = 𝐽𝑗 = 𝐽)
37 oveq1 6800 . . . . . . . . . 10 (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
3835, 36, 37ifbieq12d 4252 . . . . . . . . 9 (𝑗 = 𝐽 → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)))
3938opeq2d 4546 . . . . . . . 8 (𝑗 = 𝐽 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ = ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩)
40 opex 5060 . . . . . . . 8 ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ ∈ V
4134, 39, 20, 40ovmpt2 6943 . . . . . . 7 ((𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ) → (𝐼(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)𝐽) = ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩)
4217, 41syl 17 . . . . . 6 (𝜑 → (𝐼(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)𝐽) = ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩)
43 smatlem.1 . . . . . . 7 (𝜑 → if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)) = 𝑋)
44 smatlem.2 . . . . . . 7 (𝜑 → if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)) = 𝑌)
4543, 44opeq12d 4547 . . . . . 6 (𝜑 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ = ⟨𝑋, 𝑌⟩)
4642, 45eqtrd 2805 . . . . 5 (𝜑 → (𝐼(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)𝐽) = ⟨𝑋, 𝑌⟩)
4729, 46syl5eqr 2819 . . . 4 (𝜑 → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩) = ⟨𝑋, 𝑌⟩)
4847fveq2d 6336 . . 3 (𝜑 → (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)) = (𝐴‘⟨𝑋, 𝑌⟩))
49 df-ov 6796 . . 3 (𝑋𝐴𝑌) = (𝐴‘⟨𝑋, 𝑌⟩)
5048, 49syl6eqr 2823 . 2 (𝜑 → (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)) = (𝑋𝐴𝑌))
5128, 50eqtrd 2805 1 (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  ifcif 4225  cop 4322   class class class wbr 4786   × cxp 5247  dom cdm 5249  ccom 5253  Fun wfun 6025  cfv 6031  (class class class)co 6793  cmpt2 6795  𝑚 cmap 8009  1c1 10139   + caddc 10141   < clt 10276  cn 11222  ...cfz 12533  subMat1csmat 30199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-z 11580  df-uz 11889  df-fz 12534  df-smat 30200
This theorem is referenced by:  smattl  30204  smattr  30205  smatbl  30206  smatbr  30207  1smat1  30210  madjusmdetlem3  30235
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