Theorem smatlem 34165

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 13585	. . . . . . . 8 ⊢ (1...𝑀) ⊆ ℕ
3		smat.k	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (1...𝑀))
4	2, 3	sselid 3936	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
5		fz1ssnn 13585	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℕ
6		smat.l	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (1...𝑁))
7	5, 6	sselid 3936	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℕ)
8		smat.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m ((1...𝑀) × (1...𝑁))))
9		smatfval 34163	. . . . . . 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 1398	. . . . . 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 2810	. . . . 5 ⊢ (𝜑 → 𝑆 = (𝐴 ∘ (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)))
12	11	oveqd 7429	. . . 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 7415	. . . 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 2814	. . 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 5699	. . . . . 6 ⊢ (⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ) ↔ (𝐼 ∈ ℕ ∧ 𝐽 ∈ ℕ))
19	17, 18	sylibr 237	. . . . 5 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ (ℕ × ℕ))
20		eqid 2763	. . . . . 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 5447	. . . . . 6 ⊢ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩ ∈ V
22	20, 21	dmmpo 8069	. . . . 5 ⊢ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩) = (ℕ × ℕ)
23	19, 22	eleqtrrdi 2874	. . . 4 ⊢ (𝜑 → ⟨𝐼, 𝐽⟩ ∈ dom (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩))
24	20	mpofun 7536	. . . . 5 ⊢ Fun (𝑖 ∈ ℕ, 𝑗 ∈ ℕ ↦ ⟨if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)), if(𝑗 < 𝐿, 𝑗, (𝑗 + 1))⟩)
25		fvco 6981	. . . . 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 2798	. 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 7415	. . . . 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 23	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
32		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → (𝑖 + 1) = (𝐼 + 1))
33	30, 31, 32	ifbieq12d 4517	. . . . . . . . 9 ⊢ (𝑖 = 𝐼 → if(𝑖 < 𝐾, 𝑖, (𝑖 + 1)) = if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)))
34	33	opeq1d 4845	. . . . . . . 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 23	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → 𝑗 = 𝐽)
37		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → (𝑗 + 1) = (𝐽 + 1))
38	35, 36, 37	ifbieq12d 4517	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → if(𝑗 < 𝐿, 𝑗, (𝑗 + 1)) = if(𝐽 < 𝐿, 𝐽, (𝐽 + 1)))
39	38	opeq2d 4846	. . . . . . . 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 5447	. . . . . . . 8 ⊢ ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ ∈ V
41	34, 39, 20, 40	ovmpo 7572	. . . . . . 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 4847	. . . . . 6 ⊢ (𝜑 → ⟨if(𝐼 < 𝐾, 𝐼, (𝐼 + 1)), if(𝐽 < 𝐿, 𝐽, (𝐽 + 1))⟩ = ⟨𝑋, 𝑌⟩)
46	42, 45	eqtrd 2798	. . . . 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 2812	. . . 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 6887	. . 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 7415	. . 3 ⊢ (𝑋𝐴𝑌) = (𝐴‘⟨𝑋, 𝑌⟩)
50	48, 49	eqtr4di 2816	. 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 2798	1 ⊢ (𝜑 → (𝐼𝑆𝐽) = (𝑋𝐴𝑌))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ifcif 4488  ⟨cop 4596   class class class wbr 5110   × cxp 5661  dom cdm 5663   ∘ ccom 5667  Fun wfun 6532  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8825  1c1 11102   + caddc 11104   < clt 11244  ℕcn 12234  ...cfz 13536  subMat1csmat 34161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-z 12593  df-uz 12864  df-fz 13537  df-smat 34162

This theorem is referenced by:  smattl  34166  smattr  34167  smatbl  34168  smatbr  34169  1smat1  34172  madjusmdetlem3  34197