Theorem smatlem 33828

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 13572	. . . . . . . 8 ⊢ (1...𝑀) ⊆ ℕ
3		smat.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
4	2, 3	sselid 3956	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
5		fz1ssnn 13572	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
6		smat.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
7	5, 6	sselid 3956	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℕ)
8		smat.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
9		smatfval 33826	. . . . . . 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 2782	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
12	11	oveqd 7422	. . . 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 7408	. . . 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 2786	. . 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 5690	. . . . . 6 ⊢ (⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ) ↔ (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
19	17, 18	sylibr 234	. . . . 5 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ))
20		eqid 2735	. . . . . 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 5439	. . . . . 6 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
22	20, 21	dmmpo 8070	. . . . 5 ⊢ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (ℕ × ℕ)
23	19, 22	eleqtrrdi 2845	. . . 4 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
24	20	mpofun 7531	. . . . 5 ⊢ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
25		fvco 6977	. . . . 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 2770	. 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 7408	. . . . 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 5122	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 < 𝐾 ↔ 𝐼 < 𝐾))
31		id 22	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
32		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
33	30, 31, 32	ifbieq12d 4529	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)))
34	33	opeq1d 4855	. . . . . . . 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 5122	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 < 𝐿 ↔ 𝐽 < 𝐿))
36		id 22	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
37		oveq1 7412	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
38	35, 36, 37	ifbieq12d 4529	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)))
39	38	opeq2d 4856	. . . . . . . 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 5439	. . . . . . . 8 ⊢ ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ ∈ V
41	34, 39, 20, 40	ovmpo 7567	. . . . . . 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 4857	. . . . . 6 ⊢ (𝜑 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ = ⟨𝑋, 𝑌⟩)
46	42, 45	eqtrd 2770	. . . . 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 2784	. . . 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 6880	. . 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 7408	. . 3 ⊢ (𝑋𝐴𝑌) = (𝐴‘⟨𝑋, 𝑌⟩)
50	48, 49	eqtr4di 2788	. 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 2770	1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ifcif 4500  ⟨cop 4607   class class class wbr 5119   × cxp 5652  dom cdm 5654   ∘ ccom 5658  Fun wfun 6525  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8840  1c1 11130   + caddc 11132   < clt 11269  ℕcn 12240  ...cfz 13524  subMat1csmat 33824

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-z 12589  df-uz 12853  df-fz 13525  df-smat 33825

This theorem is referenced by:  smattl  33829  smattr  33830  smatbl  33831  smatbr  33832  1smat1  33835  madjusmdetlem3  33860