Theorem smatlem 34104

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 13574	. . . . . . . 8 ⊢ (1...𝑀) ⊆ ℕ
3		smat.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
4	2, 3	sselid 3937	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
5		fz1ssnn 13574	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
6		smat.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
7	5, 6	sselid 3937	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℕ)
8		smat.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
9		smatfval 34102	. . . . . . 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 1394	. . . . . 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 2812	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
12	11	oveqd 7417	. . . 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 7403	. . . 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 2816	. . 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 520	. . . . . 6 ⊢ (𝜑 → (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
18		opelxp 5688	. . . . . 6 ⊢ (⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ) ↔ (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
19	17, 18	sylibr 237	. . . . 5 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ))
20		eqid 2765	. . . . . 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 5436	. . . . . 6 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
22	20, 21	dmmpo 8056	. . . . 5 ⊢ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (ℕ × ℕ)
23	19, 22	eleqtrrdi 2876	. . . 4 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
24	20	mpofun 7524	. . . . 5 ⊢ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
25		fvco 6969	. . . . 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 702	. . . 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 18	. . 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 2800	. 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 7403	. . . . 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 5108	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 < 𝐾 ↔ 𝐼 < 𝐾))
31		id 23	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
32		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
33	30, 31, 32	ifbieq12d 4512	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)))
34	33	opeq1d 4840	. . . . . . . 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 5108	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 < 𝐿 ↔ 𝐽 < 𝐿))
36		id 23	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
37		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
38	35, 36, 37	ifbieq12d 4512	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)))
39	38	opeq2d 4841	. . . . . . . 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 5436	. . . . . . . 8 ⊢ ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ ∈ V
41	34, 39, 20, 40	ovmpo 7560	. . . . . . 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 18	. . . . . 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 4842	. . . . . 6 ⊢ (𝜑 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ = ⟨𝑋, 𝑌⟩)
46	42, 45	eqtrd 2800	. . . . 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 2814	. . . 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 6875	. . 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 7403	. . 3 ⊢ (𝑋𝐴𝑌) = (𝐴‘⟨𝑋, 𝑌⟩)
50	48, 49	eqtr4di 2818	. 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 2800	1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ifcif 4483  ⟨cop 4591   class class class wbr 5105   × cxp 5650  dom cdm 5652   ∘ ccom 5656  Fun wfun 6519  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   ↑_m cmap 8812  1c1 11089   + caddc 11091   < clt 11231  ℕcn 12224  ...cfz 13526  subMat1csmat 34100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-z 12583  df-uz 12854  df-fz 13527  df-smat 34101

This theorem is referenced by:  smattl  34105  smattr  34106  smatbl  34107  smatbr  34108  1smat1  34111  madjusmdetlem3  34136