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Theorem maducoeval2 22601

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 2826	. . . . . . . 8 ⊢ (𝑚 = ∅ → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ ∅))
2	1	ifbid 4505	. . . . . . 7 ⊢ (𝑚 = ∅ → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
3	2	ifeq2d 4502	. . . . . 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 7449	. . . . 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 6848	. . . 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 2826	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ 𝑛))
8	7	ifbid 4505	. . . . . . 7 ⊢ (𝑚 = 𝑛 → if(𝑘 ∈ 𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ 𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
9	8	ifeq2d 4502	. . . . . 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 7449	. . . . 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 6848	. . . 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 2826	. . . . . . . 8 ⊢ (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑛 ∪ {𝑟})))
14	13	ifbid 4505	. . . . . . 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 4502	. . . . . 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 7449	. . . . 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 6848	. . . 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 2826	. . . . . . . 8 ⊢ (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘 ∈ 𝑚 ↔ 𝑘 ∈ (𝑁 ∖ {𝐻})))
20	19	ifbid 4505	. . . . . . 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 4502	. . . . . 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 7449	. . . . 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 6848	. . . 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 22600	. . . . 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 1178	. . . 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 4292	. . . . . . . 8 ⊢ ¬ 𝑘 ∈ ∅
34		iffalse 4490	. . . . . . . 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 4502	. . . . . 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 7448	. . . . 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 6847	. . . 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 2790	. . 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 1226	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑅 ∈ CRing)
44		simp1r 1200	. . . . . . . . 9 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑀 ∈ 𝐵)
45	25, 28	matrcl 22373	. . . . . . . . . 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 1199	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → 𝑅 ∈ CRing)
50	49	ad2antrr 727	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙 ∈ 𝑁) → 𝑅 ∈ CRing)
51		crngring 20197	. . . . . . . . . 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 20219	. . . . . . . . 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 1227	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀 ∈ 𝐵)
56	25, 40, 28	matbas2i 22383	. . . . . . . . . . 11 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
57		elmapi 8800	. . . . . . . . . . 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 4085	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
61	60	ad2antll 730	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
62	61	eldifad 3915	. . . . . . . . . 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 7542	. . . . . . . 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 4528	. . . . . . 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 20217	. . . . . . . . 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 4528	. . . . . . 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 7541	. . . . . . . . . 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 4528	. . . . . . . 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 4528	. . . . . . 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 1194	. . . . . . . 8 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐼 ∈ 𝑁)
76	58, 62, 75	fovcdmd 7542	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
77		simpl3 1195	. . . . . . 7 ⊢ ((((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐻 ∈ 𝑁)
78		eldifsni 4748	. . . . . . . 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 22571	. . . . . 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 4534	. . . . . . . . . . . . . . . . 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 7469	. . . . . . . . . . . . . . . 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 20222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 1 ) = (𝑟𝑀𝐼))
87	52, 85, 86	syl2anc 585	. . . . . . . . . . . . . . . . . . . 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 7378	. . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . 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 4513	. . . . . . . . . . . . . . . . 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 20246	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(._r‘𝑅) 0 ) = 0 )
94	52, 85, 93	syl2anc 585	. . . . . . . . . . . . . . . . . 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 4502	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . 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 7388	. . . . . . . . . . . . . 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 20195	. . . . . . . . . . . . . . . 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 22597	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 4487	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 7377	. . . . . . . . . . . . 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 4487	. . . . . . . . . . 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 2938	. . . . . . . . . . . . . . 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 4492	. . . . . . . . . . 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 4086	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟 ∈ 𝑛)
126	125	ad2antll 730	. . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . 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 4492	. . . . . . . . . . 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 2772	. . . . . . . . . 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 2782	. . . . . . . . 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 4490	. . . . . . . . . 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 7447	. . . . . . 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 6848	. . . . . 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 2996	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝐻 → (𝑟 ≠ 𝑘 ↔ 𝑟 ≠ 𝐻))
141	140	biimparc 479	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑟 ≠ 𝑘)
142	141	necomd 2988	. . . . . . . . . . . . 13 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → 𝑘 ≠ 𝑟)
143	142	neneqd 2938	. . . . . . . . . . . 12 ⊢ ((𝑟 ≠ 𝐻 ∧ 𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144	143	iffalsed 4492	. . . . . . . . . . 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 4487	. . . . . . . . . . . . 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 4502	. . . . . . . . . . 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 4487	. . . . . . . . . . . 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 2782	. . . . . . . . . 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 4502	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152		vsnid 4622	. . . . . . . . . . . . . . . . 17 ⊢ 𝑟 ∈ {𝑟}
153		elun2 4137	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154	152, 153	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 𝑟 ∈ (𝑛 ∪ {𝑟})
155		eleq1w 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156	154, 155	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑟 → 𝑘 ∈ (𝑛 ∪ {𝑟}))
157	156	iftrued 4489	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158		iftrue 4487	. . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . 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 4490	. . . . . . . . . . . . . 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 4107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘 ∈ 𝑛 ∨ 𝑘 ∈ {𝑟}))
166		velsn 4598	. . . . . . . . . . . . . . . . . 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 4505	. . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . 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 4490	. . . . . . . . . . . . 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 4502	. . . . . . . . . . . 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 4490	. . . . . . . . . . . 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 2782	. . . . . . . . . 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 7449	. . . . . . . 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 6848	. . . . . . 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 2780	. . . . 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 4090	. . . 4 ⊢ (𝑁 ∖ {𝐻}) ⊆ 𝑁
188		ssfi 9111	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝐻}) ⊆ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
189	47, 187, 188	sylancl 587	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐻 ∈ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
190	6, 12, 18, 24, 39, 186, 189	findcard2d 9105	. 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 4505	. . . . . . 7 ⊢ (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼 ∧ 𝑘 = 𝐻), 1 , 0 ))
193		iftrue 4487	. . . . . . 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 4487	. . . . . . . 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 2782	. . . . . 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 4490	. . . . . . 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 2941	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 𝐻 → 𝑘 ≠ 𝐻)
201	200	anim2i 618	. . . . . . . . 9 ⊢ ((𝑘 ∈ 𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
202	201	adantlr 716	. . . . . . . 8 ⊢ (((𝑘 ∈ 𝑁 ∧ 𝑙 ∈ 𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘 ∈ 𝑁 ∧ 𝑘 ≠ 𝐻))
203		eldifsn 4744	. . . . . . . 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 4489	. . . . . 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 4492	. . . . . . . . 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 4506	. . . . . . 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 2776	. . . . 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 7448	. . 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 6847	. 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 2788	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 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 ∅c0 4287 ifcif 4481 {csn 4582 × cxp 5632 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ∈ cmpo 7372 ↑_m cmap 8777 Fincfn 8897 Basecbs 17150 +_gcplusg 17191 ._rcmulr 17192 0_gc0g 17373 Mndcmnd 18673 1_rcur 20133 Ringcrg 20185 CRingccrg 20186 Mat cmat 22368 maDet cmdat 22545 maAdju cmadu 22593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-addf 11119 ax-mulf 11120

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-tpos 8180 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-sup 9359 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-xnn0 12489 df-z 12503 df-dec 12622 df-uz 12766 df-rp 12920 df-fz 13438 df-fzo 13585 df-seq 13939 df-exp 13999 df-hash 14268 df-word 14451 df-lsw 14500 df-concat 14508 df-s1 14534 df-substr 14579 df-pfx 14609 df-splice 14687 df-reverse 14696 df-s2 14785 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-0g 17375 df-gsum 17376 df-prds 17381 df-pws 17383 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-mhm 18722 df-submnd 18723 df-efmnd 18808 df-grp 18883 df-minusg 18884 df-mulg 19015 df-subg 19070 df-ghm 19159 df-gim 19205 df-cntz 19263 df-oppg 19292 df-symg 19316 df-pmtr 19388 df-psgn 19437 df-evpm 19438 df-cmn 19728 df-abl 19729 df-mgp 20093 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20290 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-rhm 20425 df-subrng 20496 df-subrg 20520 df-drng 20681 df-sra 21142 df-rgmod 21143 df-cnfld 21327 df-zring 21419 df-zrh 21475 df-dsmm 21704 df-frlm 21719 df-mat 22369 df-mdet 22546 df-madu 22595

This theorem is referenced by: madutpos 22603

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