Theorem smatlem 33794

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	⊢ (𝜑 → 𝑀 ∈ ℕ)
smat.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
smat.k	⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
smat.l	⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
smat.a	⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
smatlem.i	⊢ (𝜑 → 𝐼 ∈ ℕ)
smatlem.j	⊢ (𝜑 → 𝐽 ∈ 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.

Step	Hyp	Ref	Expression
1		smat.s	. . . . . 6 ⊢ 𝑆 = (𝐾(subMat1‘𝐴)𝐿)
2		fz1ssnn 13523	. . . . . . . 8 ⊢ (1...𝑀) ⊆ ℕ
3		smat.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
4	2, 3	sselid 3947	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
5		fz1ssnn 13523	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
6		smat.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
7	5, 6	sselid 3947	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℕ)
8		smat.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
9		smatfval 33792	. . . . . . 7 ⊢ ((𝐾 ∈ ℕ ∧ 𝐿 ∈ ℕ ∧ 𝐴 ∈ (𝐵 ↑m ((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))⟩)))
10	4, 7, 8, 9	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐾(subMat1‘𝐴)𝐿) = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
11	1, 10	eqtrid 2777	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
12	11	oveqd 7407	. . . 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 7393	. . . 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⟩)
14	12, 13	eqtrdi 2781	. . 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 ⊢ (𝜑 → 𝐼 ∈ ℕ)
16		smatlem.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ)
17	15, 16	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
18		opelxp 5677	. . . . . 6 ⊢ (⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ) ↔ (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
19	17, 18	sylibr 234	. . . . 5 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ))
20		eqid 2730	. . . . . 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 5427	. . . . . 6 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
22	20, 21	dmmpo 8053	. . . . 5 ⊢ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (ℕ × ℕ)
23	19, 22	eleqtrrdi 2840	. . . 4 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
24	20	mpofun 7516	. . . . 5 ⊢ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
25		fvco 6962	. . . . 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⟩)))
26	24, 25	mpan 690	. . . 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⟩)))
27	23, 26	syl 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⟩)))
28	14, 27	eqtrd 2765	. 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 7393	. . . . 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 5113	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 < 𝐾 ↔ 𝐼 < 𝐾))
31		id 22	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
32		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
33	30, 31, 32	ifbieq12d 4520	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)))
34	33	opeq1d 4846	. . . . . . . 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 5113	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 < 𝐿 ↔ 𝐽 < 𝐿))
36		id 22	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
37		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
38	35, 36, 37	ifbieq12d 4520	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)))
39	38	opeq2d 4847	. . . . . . . 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 5427	. . . . . . . 8 ⊢ ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ ∈ V
41	34, 39, 20, 40	ovmpo 7552	. . . . . . 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))⟩)
42	17, 41	syl 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)
45	43, 44	opeq12d 4848	. . . . . 6 ⊢ (𝜑 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ = ⟨𝑋, 𝑌⟩)
46	42, 45	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (𝐼(𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)𝐽) = ⟨𝑋, 𝑌⟩)
47	29, 46	eqtr3id 2779	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩) = ⟨𝑋, 𝑌⟩)
48	47	fveq2d 6865	. . 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 7393	. . 3 ⊢ (𝑋𝐴𝑌) = (𝐴‘⟨𝑋, 𝑌⟩)
50	48, 49	eqtr4di 2783	. 2 ⊢ (𝜑 → (𝐴‘((𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)‘⟨𝐼, 𝐽⟩)) = (𝑋𝐴𝑌))
51	28, 50	eqtrd 2765	1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ifcif 4491  ⟨cop 4598   class class class wbr 5110   × cxp 5639  dom cdm 5641   ∘ ccom 5645  Fun wfun 6508  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ↑_m cmap 8802  1c1 11076   + caddc 11078   < clt 11215  ℕcn 12193  ...cfz 13475  subMat1csmat 33790

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-z 12537  df-uz 12801  df-fz 13476  df-smat 33791

This theorem is referenced by:  smattl  33795  smattr  33796  smatbl  33797  smatbr  33798  1smat1  33801  madjusmdetlem3  33826