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Theorem maducoeval2 22646

Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 17-Jul-2018.)

**Hypotheses**
Ref	Expression
madufval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
madufval.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
madufval.j	⊢ 𝐽 = (𝑁 maAdju 𝑅)
madufval.b	⊢ 𝐵 = (Base‘𝐴)
madufval.o	⊢ 1 = (1_r‘𝑅)
madufval.z	⊢ 0 = (0_g‘𝑅)

**Assertion**
Ref	Expression
maducoeval2	⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Distinct variable groups: 𝑘,𝑁,𝑙 𝑅,𝑘,𝑙 𝑘,𝑀,𝑙 𝑘,𝐼,𝑙 𝑘,𝐻,𝑙 𝐵,𝑘,𝑙 0 ,𝑘 1 ,𝑘 Allowed substitution hints: 𝐴(𝑘,𝑙) 𝐷(𝑘,𝑙) 1 (𝑙) 𝐽(𝑘,𝑙) 0 (𝑙)

Proof of Theorem maducoeval2 Dummy variables 𝑛 𝑟 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2830	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ∅))
2	1	ifbid 4549	. . . . . . 7 ⊢ (𝑚 = ∅ → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
3	2	ifeq2d 4546	. . . . . 6 ⊢ (𝑚 = ∅ → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
4	3	mpoeq3dv 7512	. . . . 5 ⊢ (𝑚 = ∅ → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
5	4	fveq2d 6910	. . . 4 ⊢ (𝑚 = ∅ → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
6	5	eqeq2d 2748	. . 3 ⊢ (𝑚 = ∅ → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
7		eleq2 2830	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ 𝑛))
8	7	ifbid 4549	. . . . . . 7 ⊢ (𝑚 = 𝑛 → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
9	8	ifeq2d 4546	. . . . . 6 ⊢ (𝑚 = 𝑛 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
10	9	mpoeq3dv 7512	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
11	10	fveq2d 6910	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
12	11	eqeq2d 2748	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
13		eleq2 2830	. . . . . . . 8 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑛 ∪ {𝑟})))
14	13	ifbid 4549	. . . . . . 7 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
15	14	ifeq2d 4546	. . . . . 6 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
16	15	mpoeq3dv 7512	. . . . 5 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
17	16	fveq2d 6910	. . . 4 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
18	17	eqeq2d 2748	. . 3 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
19		eleq2 2830	. . . . . . . 8 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑁 ∖ {𝐻})))
20	19	ifbid 4549	. . . . . . 7 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
21	20	ifeq2d 4546	. . . . . 6 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
22	21	mpoeq3dv 7512	. . . . 5 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
23	22	fveq2d 6910	. . . 4 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
24	23	eqeq2d 2748	. . 3 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
25		madufval.a	. . . . . 6 ⊢ 𝐴 = (𝑁 Mat 𝑅)
26		madufval.d	. . . . . 6 ⊢ 𝐷 = (𝑁 maDet 𝑅)
27		madufval.j	. . . . . 6 ⊢ 𝐽 = (𝑁 maAdju 𝑅)
28		madufval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
29		madufval.o	. . . . . 6 ⊢ 1 = (1_r‘𝑅)
30		madufval.z	. . . . . 6 ⊢ 0 = (0_g‘𝑅)
31	25, 26, 27, 28, 29, 30	maducoeval 22645	. . . . 5 ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
32	31	3adant1l 1177	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
33		noel 4338	. . . . . . . 8 ⊢ ¬ 𝑘 ∈ ∅
34		iffalse 4534	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ ∅ → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
35	33, 34	mp1i 13	. . . . . . 7 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
36	35	ifeq2d 4546	. . . . . 6 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
37	36	mpoeq3ia 7511	. . . . 5 ⊢ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
38	37	fveq2i 6909	. . . 4 ⊢ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))
39	32, 38	eqtr4di 2795	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
40		eqid 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
41		eqid 2737	. . . . . . 7 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
42		eqid 2737	. . . . . . 7 ⊢ (._r‘𝑅) = (._r‘𝑅)
43		simpl1l 1225	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑅 ∈ CRing)
44		simp1r 1199	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑀 ∈ 𝐵)
45	25, 28	matrcl 22416	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
46	45	simpld 494	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
47	44, 46	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑁 ∈ Fin)
48	47	adantr 480	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑁 ∈ Fin)
49		simp1l 1198	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑅 ∈ CRing)
50	49	ad2antrr 726	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ CRing)
51		crngring 20242	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
52	50, 51	syl 17	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Ring)
53	40, 30	ring0cl 20264	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
54	52, 53	syl 17	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
55		simpl1r 1226	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀 ∈ 𝐵)
56	25, 40, 28	matbas2i 22428	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
57		elmapi 8889	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
58	55, 56, 57	3syl 18	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
59	58	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
60		eldifi 4131	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
61	60	ad2antll 729	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
62	61	eldifad 3963	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ 𝑁)
63	62	adantr 480	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑟 ∈ 𝑁)
64		simpr 484	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑙 ∈ 𝑁)
65	59, 63, 64	fovcdmd 7605	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (𝑟𝑀𝑙) ∈ (Base‘𝑅))
66	54, 65	ifcld 4572	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) ∈ (Base‘𝑅))
67	40, 29	ringidcl 20262	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
68	52, 67	syl 17	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 1 ∈ (Base‘𝑅))
69	68, 54	ifcld 4572	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅))
70	54	3adant2 1132	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → 0 ∈ (Base‘𝑅))
71	58	fovcdmda 7604	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ (𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
72	71	3impb 1115	. . . . . . . . 9 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
73	70, 72	ifcld 4572	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
74	73, 72	ifcld 4572	. . . . . . 7 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
75		simpl2 1193	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐼 ∈ 𝑁)
76	58, 62, 75	fovcdmd 7605	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
77		simpl3 1194	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐻 ∈ 𝑁)
78		eldifsni 4790	. . . . . . . 8 ⊢ (𝑟 ∈ (𝑁 ∖ {𝐻}) → 𝑟 ≠ 𝐻)
79	61, 78	syl 17	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ≠ 𝐻)
80	26, 40, 41, 42, 43, 48, 66, 69, 74, 76, 62, 77, 79	mdetero 22616	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
81		ifnot 4578	. . . . . . . . . . . . . . . . 17 ⊢ if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))
82	81	eqcomi 2746	. . . . . . . . . . . . . . . 16 ⊢ if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )
83	82	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
84		ovif2 7532	. . . . . . . . . . . . . . . 16 ⊢ ((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 ))
85	76	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
86	40, 42, 29	ringridm 20267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
87	52, 85, 86	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
88	87	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
89		oveq2 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝐼 → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
90	89	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
91	88, 90	eqtr4d 2780	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝑙))
92	91	ifeq1da 4557	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )))
93	40, 42, 30	ringrz 20291	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 0 ) = 0 )
94	52, 85, 93	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅) 0 ) = 0 )
95	94	ifeq2d 4546	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
96	92, 95	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(._r‘𝑅) 1 ), ((𝑟𝑀𝐼)(._r‘𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
97	84, 96	eqtrid 2789	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
98	83, 97	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
99		ringmnd 20240	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
100	52, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ Mnd)
101		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼)
102		imnan 399	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼) ↔ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼))
103	101, 102	mpbi 230	. . . . . . . . . . . . . . . 16 ⊢ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼)
104	103	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼))
105	40, 30, 41	mndifsplit 22642	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Mnd ∧ (𝑟𝑀𝑙) ∈ (Base‘𝑅) ∧ ¬ (¬ 𝑙 = 𝐼 ∧ 𝑙 = 𝐼)) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
106	100, 65, 104, 105	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+_g‘𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
107		pm2.1 897	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼)
108		iftrue 4531	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
109	107, 108	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → if((¬ 𝑙 = 𝐼 ∨ 𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
110	98, 106, 109	3eqtr2d 2783	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
111	110	3adant2 1132	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
112		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑟 → (𝑘𝑀𝑙) = (𝑟𝑀𝑙))
113	112	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑟 → ((if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙) ↔ (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙)))
114	111, 113	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙)))
115	114	imp 406	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙))
116		iftrue 4531	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
117	116	adantl 481	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
118	79	neneqd 2945	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 = 𝐻)
119	118	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → ¬ 𝑟 = 𝐻)
120		eqeq1 2741	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → (𝑘 = 𝐻 ↔ 𝑟 = 𝐻))
121	120	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → (¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻))
122	119, 121	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 = 𝐻))
123	122	imp 406	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 = 𝐻)
124	123	iffalsed 4536	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
125		eldifn 4132	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟 ∈ 𝑛)
126	125	ad2antll 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 ∈ 𝑛)
127	126	3ad2ant1 1134	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → ¬ 𝑟 ∈ 𝑛)
128		eleq1w 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ 𝑟 ∈ 𝑛))
129	128	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → (¬ 𝑘 ∈ 𝑛 ↔ ¬ 𝑟 ∈ 𝑛))
130	127, 129	syl5ibrcom 247	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 ∈ 𝑛))
131	130	imp 406	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 ∈ 𝑛)
132	131	iffalsed 4536	. . . . . . . . . . 11 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
133	124, 132	eqtrd 2777	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = (𝑘𝑀𝑙))
134	115, 117, 133	3eqtr4d 2787	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
135		iffalse 4534	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
136	135	adantl 481	. . . . . . . . 9 ⊢ ((((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
137	134, 136	pm2.61dan 813	. . . . . . . 8 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
138	137	mpoeq3dva 7510	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
139	138	fveq2d 6910	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+_g‘𝑅)((𝑟𝑀𝐼)(._r‘𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
140		neeq2 3004	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐻 → (𝑟 ≠ 𝑘 ↔ 𝑟 ≠ 𝐻))
141	140	biimparc 479	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑟 ≠ 𝑘)
142	141	necomd 2996	. . . . . . . . . . . . 13 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑘 ≠ 𝑟)
143	142	neneqd 2945	. . . . . . . . . . . 12 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144	143	iffalsed 4536	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, 1 , 0 ))
145		iftrue 4531	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
146	145	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
147	146	ifeq2d 4546	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )))
148		iftrue 4531	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
149	148	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
150	144, 147, 149	3eqtr4d 2787	. . . . . . . . . 10 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
151	112	ifeq2d 4546	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152		vsnid 4663	. . . . . . . . . . . . . . . . 17 ⊢ 𝑟 ∈ {𝑟}
153		elun2 4183	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154	152, 153	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 𝑟 ∈ (𝑛 ∪ {𝑟})
155		eleq1w 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156	154, 155	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → 𝑘 ∈ (𝑛 ∪ {𝑟}))
157	156	iftrued 4533	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158		iftrue 4531	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
159	151, 157, 158	3eqtr4rd 2788	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
160	159	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
161		iffalse 4534	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
162		orc 868	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑛 → (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟))
163		orel2 891	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑘 = 𝑟 → ((𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟) → 𝑘 ∈ 𝑛))
164	162, 163	impbid2 226	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟)))
165		elun 4153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 ∈ {𝑟}))
166		velsn 4642	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ {𝑟} ↔ 𝑘 = 𝑟)
167	166	orbi2i 913	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ 𝑛 ∨ 𝑘 ∈ {𝑟}) ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟))
168	165, 167	bitr2i 276	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝑛 ∨ 𝑘 = 𝑟) ↔ 𝑘 ∈ (𝑛 ∪ {𝑟}))
169	164, 168	bitrdi 287	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑘 = 𝑟 → (𝑘 ∈ 𝑛 ↔ 𝑘 ∈ (𝑛 ∪ {𝑟})))
170	169	ifbid 4549	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
171	161, 170	eqtrd 2777	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
172	171	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
173	160, 172	pm2.61dan 813	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
174		iffalse 4534	. . . . . . . . . . . . 13 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
175	174	ifeq2d 4546	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
176	175	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
177		iffalse 4534	. . . . . . . . . . . 12 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
178	177	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
179	173, 176, 178	3eqtr4d 2787	. . . . . . . . . 10 ⊢ ((𝑟 ≠ 𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
180	150, 179	pm2.61dan 813	. . . . . . . . 9 ⊢ (𝑟 ≠ 𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
181	180	mpoeq3dv 7512	. . . . . . . 8 ⊢ (𝑟 ≠ 𝐻 → (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
182	181	fveq2d 6910	. . . . . . 7 ⊢ (𝑟 ≠ 𝐻 → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
183	79, 182	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
184	80, 139, 183	3eqtr3d 2785	. . . . 5 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
185	184	eqeq2d 2748	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
186	185	biimpd 229	. . 3 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
187		difss 4136	. . . 4 ⊢ (𝑁 ∖ {𝐻}) ⊆ 𝑁
188		ssfi 9213	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝐻}) ⊆ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
189	47, 187, 188	sylancl 586	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
190	6, 12, 18, 24, 39, 186, 189	findcard2d 9206	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
191		iba 527	. . . . . . . 8 ⊢ (𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑙 = 𝐼 ∧ 𝑘 = 𝐻)))
192	191	ifbid 4549	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
193		iftrue 4531	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
194		iftrue 4531	. . . . . . . 8 ⊢ ((𝑘 = 𝐻 ∨ 𝑙 = 𝐼) → if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
195	194	orcs 876	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
196	192, 193, 195	3eqtr4d 2787	. . . . . 6 ⊢ (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
197	196	adantl 481	. . . . 5 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
198		iffalse 4534	. . . . . . 7 ⊢ (¬ 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
199	198	adantl 481	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
200		neqne 2948	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝐻 → 𝑘 ≠ 𝐻)
201	200	anim2i 617	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
202	201	adantlr 715	. . . . . . . 8 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
203		eldifsn 4786	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑁 ∖ {𝐻}) ↔ (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
204	202, 203	sylibr 234	. . . . . . 7 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → 𝑘 ∈ (𝑁 ∖ {𝐻}))
205	204	iftrued 4533	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
206		biorf 937	. . . . . . . 8 ⊢ (¬ 𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑘 = 𝐻 ∨ 𝑙 = 𝐼)))
207		id 22	. . . . . . . . . . 11 ⊢ (¬ 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻)
208	207	intnand 488	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝐻 → ¬ (𝑙 = 𝐼 ∧ 𝑘 = 𝐻))
209	208	iffalsed 4536	. . . . . . . . 9 ⊢ (¬ 𝑘 = 𝐻 → if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ) = 0 )
210	209	eqcomd 2743	. . . . . . . 8 ⊢ (¬ 𝑘 = 𝐻 → 0 = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
211	206, 210	ifbieq1d 4550	. . . . . . 7 ⊢ (¬ 𝑘 = 𝐻 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
212	211	adantl 481	. . . . . 6 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
213	199, 205, 212	3eqtrd 2781	. . . . 5 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
214	197, 213	pm2.61dan 813	. . . 4 ⊢ ((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
215	214	mpoeq3ia 7511	. . 3 ⊢ (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
216	215	fveq2i 6909	. 2 ⊢ (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙))))
217	190, 216	eqtrdi 2793	1 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝐼(𝐽‘𝑀)𝐻) = (𝐷‘(𝑘 ∈ 𝑁, 𝑙 ∈ 𝑁 ↦ if((𝑘 = 𝐻 ∨ 𝑙 = 𝐼), if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 ∅c0 4333 ifcif 4525 {csn 4626 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑_m cmap 8866 Fincfn 8985 Basecbs 17247 +_gcplusg 17297 ._rcmulr 17298 0_gc0g 17484 Mndcmnd 18747 1_rcur 20178 Ringcrg 20230 CRingccrg 20231 Mat cmat 22411 maDet cmdat 22590 maAdju cmadu 22638

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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-word 14553 df-lsw 14601 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-reverse 14797 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-efmnd 18882 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 df-ghm 19231 df-gim 19277 df-cntz 19335 df-oppg 19364 df-symg 19387 df-pmtr 19460 df-psgn 19509 df-evpm 19510 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-zrh 21514 df-dsmm 21752 df-frlm 21767 df-mat 22412 df-mdet 22591 df-madu 22640

This theorem is referenced by: madutpos 22648

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