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Theorem maducoeval2 21798
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 = (1r𝑅)
madufval.z 0 = (0g𝑅)
Assertion
Ref Expression
maducoeval2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 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.
StepHypRef Expression
1 eleq2 2828 . . . . . . . 8 (𝑚 = ∅ → (𝑘𝑚𝑘 ∈ ∅))
21ifbid 4483 . . . . . . 7 (𝑚 = ∅ → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
32ifeq2d 4480 . . . . . 6 (𝑚 = ∅ → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
43mpoeq3dv 7363 . . . . 5 (𝑚 = ∅ → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
54fveq2d 6787 . . . 4 (𝑚 = ∅ → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
65eqeq2d 2750 . . 3 (𝑚 = ∅ → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
7 eleq2 2828 . . . . . . . 8 (𝑚 = 𝑛 → (𝑘𝑚𝑘𝑛))
87ifbid 4483 . . . . . . 7 (𝑚 = 𝑛 → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
98ifeq2d 4480 . . . . . 6 (𝑚 = 𝑛 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
109mpoeq3dv 7363 . . . . 5 (𝑚 = 𝑛 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
1110fveq2d 6787 . . . 4 (𝑚 = 𝑛 → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
1211eqeq2d 2750 . . 3 (𝑚 = 𝑛 → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
13 eleq2 2828 . . . . . . . 8 (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘𝑚𝑘 ∈ (𝑛 ∪ {𝑟})))
1413ifbid 4483 . . . . . . 7 (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
1514ifeq2d 4480 . . . . . 6 (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
1615mpoeq3dv 7363 . . . . 5 (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
1716fveq2d 6787 . . . 4 (𝑚 = (𝑛 ∪ {𝑟}) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
1817eqeq2d 2750 . . 3 (𝑚 = (𝑛 ∪ {𝑟}) → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
19 eleq2 2828 . . . . . . . 8 (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘𝑚𝑘 ∈ (𝑁 ∖ {𝐻})))
2019ifbid 4483 . . . . . . 7 (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
2120ifeq2d 4480 . . . . . 6 (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
2221mpoeq3dv 7363 . . . . 5 (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
2322fveq2d 6787 . . . 4 (𝑚 = (𝑁 ∖ {𝐻}) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
2423eqeq2d 2750 . . 3 (𝑚 = (𝑁 ∖ {𝐻}) → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ 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 = (1r𝑅)
30 madufval.z . . . . . 6 0 = (0g𝑅)
3125, 26, 27, 28, 29, 30maducoeval 21797 . . . . 5 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
32313adant1l 1175 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
33 noel 4265 . . . . . . . 8 ¬ 𝑘 ∈ ∅
34 iffalse 4469 . . . . . . . 8 𝑘 ∈ ∅ → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
3533, 34mp1i 13 . . . . . . 7 ((𝑘𝑁𝑙𝑁) → if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
3635ifeq2d 4480 . . . . . 6 ((𝑘𝑁𝑙𝑁) → 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)))
3736mpoeq3ia 7362 . . . . 5 (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))
3837fveq2i 6786 . . . 4 (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙))))
3932, 38eqtr4di 2797 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
40 eqid 2739 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
41 eqid 2739 . . . . . . 7 (+g𝑅) = (+g𝑅)
42 eqid 2739 . . . . . . 7 (.r𝑅) = (.r𝑅)
43 simpl1l 1223 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑅 ∈ CRing)
44 simp1r 1197 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑀𝐵)
4525, 28matrcl 21568 . . . . . . . . . 10 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4645simpld 495 . . . . . . . . 9 (𝑀𝐵𝑁 ∈ Fin)
4744, 46syl 17 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑁 ∈ Fin)
4847adantr 481 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑁 ∈ Fin)
49 simp1l 1196 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑅 ∈ CRing)
5049ad2antrr 723 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑅 ∈ CRing)
51 crngring 19804 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
5250, 51syl 17 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑅 ∈ Ring)
5340, 30ring0cl 19817 . . . . . . . . 9 (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅))
5452, 53syl 17 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 0 ∈ (Base‘𝑅))
55 simpl1r 1224 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀𝐵)
5625, 40, 28matbas2i 21580 . . . . . . . . . . 11 (𝑀𝐵𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
57 elmapi 8646 . . . . . . . . . . 11 (𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
5855, 56, 573syl 18 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
5958adantr 481 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
60 eldifi 4062 . . . . . . . . . . . 12 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6160ad2antll 726 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6261eldifad 3900 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟𝑁)
6362adantr 481 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑟𝑁)
64 simpr 485 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑙𝑁)
6559, 63, 64fovrnd 7453 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (𝑟𝑀𝑙) ∈ (Base‘𝑅))
6654, 65ifcld 4506 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) ∈ (Base‘𝑅))
6740, 29ringidcl 19816 . . . . . . . . 9 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
6852, 67syl 17 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 1 ∈ (Base‘𝑅))
6968, 54ifcld 4506 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅))
70543adant2 1130 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → 0 ∈ (Base‘𝑅))
7158fovrnda 7452 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ (𝑘𝑁𝑙𝑁)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
72713impb 1114 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
7370, 72ifcld 4506 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
7473, 72ifcld 4506 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
75 simpl2 1191 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐼𝑁)
7658, 62, 75fovrnd 7453 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
77 simpl3 1192 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐻𝑁)
78 eldifsni 4724 . . . . . . . 8 (𝑟 ∈ (𝑁 ∖ {𝐻}) → 𝑟𝐻)
7961, 78syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟𝐻)
8026, 40, 41, 42, 43, 48, 66, 69, 74, 76, 62, 77, 79mdetero 21768 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
81 ifnot 4512 . . . . . . . . . . . . . . . . 17 if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))
8281eqcomi 2748 . . . . . . . . . . . . . . . 16 if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )
8382a1i 11 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) = if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
84 ovif2 7382 . . . . . . . . . . . . . . . 16 ((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(.r𝑅) 1 ), ((𝑟𝑀𝐼)(.r𝑅) 0 ))
8576adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
8640, 42, 29ringridm 19820 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝐼))
8752, 85, 86syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝐼))
8887adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝐼))
89 oveq2 7292 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝐼 → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
9089adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
9188, 90eqtr4d 2782 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝑙))
9291ifeq1da 4491 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(.r𝑅) 1 ), ((𝑟𝑀𝐼)(.r𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(.r𝑅) 0 )))
9340, 42, 30ringrz 19836 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(.r𝑅) 0 ) = 0 )
9452, 85, 93syl2anc 584 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → ((𝑟𝑀𝐼)(.r𝑅) 0 ) = 0 )
9594ifeq2d 4480 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, (𝑟𝑀𝑙), ((𝑟𝑀𝐼)(.r𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
9692, 95eqtrd 2779 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(.r𝑅) 1 ), ((𝑟𝑀𝐼)(.r𝑅) 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
9784, 96eqtrid 2791 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → ((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ))
9883, 97oveq12d 7302 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+g𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
99 ringmnd 19802 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
10052, 99syl 17 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑅 ∈ Mnd)
101 id 22 . . . . . . . . . . . . . . . . 17 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼)
102 imnan 400 . . . . . . . . . . . . . . . . 17 ((¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼) ↔ ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼))
103101, 102mpbi 229 . . . . . . . . . . . . . . . 16 ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼)
104103a1i 11 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼))
10540, 30, 41mndifsplit 21794 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Mnd ∧ (𝑟𝑀𝑙) ∈ (Base‘𝑅) ∧ ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼)) → if((¬ 𝑙 = 𝐼𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+g𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
106100, 65, 104, 105syl3anc 1370 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if((¬ 𝑙 = 𝐼𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )(+g𝑅)if(𝑙 = 𝐼, (𝑟𝑀𝑙), 0 )))
107 pm2.1 894 . . . . . . . . . . . . . . 15 𝑙 = 𝐼𝑙 = 𝐼)
108 iftrue 4466 . . . . . . . . . . . . . . 15 ((¬ 𝑙 = 𝐼𝑙 = 𝐼) → if((¬ 𝑙 = 𝐼𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
109107, 108mp1i 13 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if((¬ 𝑙 = 𝐼𝑙 = 𝐼), (𝑟𝑀𝑙), 0 ) = (𝑟𝑀𝑙))
11098, 106, 1093eqtr2d 2785 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
1111103adant2 1130 . . . . . . . . . . . 12 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙))
112 oveq1 7291 . . . . . . . . . . . . 13 (𝑘 = 𝑟 → (𝑘𝑀𝑙) = (𝑟𝑀𝑙))
113112eqeq2d 2750 . . . . . . . . . . . 12 (𝑘 = 𝑟 → ((if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙) ↔ (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑟𝑀𝑙)))
114111, 113syl5ibrcom 246 . . . . . . . . . . 11 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙)))
115114imp 407 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))) = (𝑘𝑀𝑙))
116 iftrue 4466 . . . . . . . . . . 11 (𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
117116adantl 482 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))))
11879neneqd 2949 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 = 𝐻)
1191183ad2ant1 1132 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → ¬ 𝑟 = 𝐻)
120 eqeq1 2743 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘 = 𝐻𝑟 = 𝐻))
121120notbid 318 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻))
122119, 121syl5ibrcom 246 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 = 𝐻))
123122imp 407 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 = 𝐻)
124123iffalsed 4471 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
125 eldifn 4063 . . . . . . . . . . . . . . . 16 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟𝑛)
126125ad2antll 726 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟𝑛)
1271263ad2ant1 1132 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → ¬ 𝑟𝑛)
128 eleq1w 2822 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘𝑛𝑟𝑛))
129128notbid 318 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘𝑛 ↔ ¬ 𝑟𝑛))
130127, 129syl5ibrcom 246 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘𝑛))
131130imp 407 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘𝑛)
132131iffalsed 4471 . . . . . . . . . . 11 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = (𝑘𝑀𝑙))
133124, 132eqtrd 2779 . . . . . . . . . 10 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = (𝑘𝑀𝑙))
134115, 117, 1333eqtr4d 2789 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
135 iffalse 4469 . . . . . . . . . 10 𝑘 = 𝑟 → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
136135adantl 482 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
137134, 136pm2.61dan 810 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
138137mpoeq3dva 7361 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
139138fveq2d 6787 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, (if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))(+g𝑅)((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 ))), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
140 neeq2 3008 . . . . . . . . . . . . . . 15 (𝑘 = 𝐻 → (𝑟𝑘𝑟𝐻))
141140biimparc 480 . . . . . . . . . . . . . 14 ((𝑟𝐻𝑘 = 𝐻) → 𝑟𝑘)
142141necomd 3000 . . . . . . . . . . . . 13 ((𝑟𝐻𝑘 = 𝐻) → 𝑘𝑟)
143142neneqd 2949 . . . . . . . . . . . 12 ((𝑟𝐻𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144143iffalsed 4471 . . . . . . . . . . 11 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, 1 , 0 ))
145 iftrue 4466 . . . . . . . . . . . . 13 (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
146145adantl 482 . . . . . . . . . . . 12 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
147146ifeq2d 4480 . . . . . . . . . . 11 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )))
148 iftrue 4466 . . . . . . . . . . . 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 ))
149148adantl 482 . . . . . . . . . . 11 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 1 , 0 ))
150144, 147, 1493eqtr4d 2789 . . . . . . . . . 10 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
151112ifeq2d 4480 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152 vsnid 4599 . . . . . . . . . . . . . . . . 17 𝑟 ∈ {𝑟}
153 elun2 4112 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16 𝑟 ∈ (𝑛 ∪ {𝑟})
155 eleq1w 2822 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156154, 155mpbiri 257 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟𝑘 ∈ (𝑛 ∪ {𝑟}))
157156iftrued 4468 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158 iftrue 4466 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
159151, 157, 1583eqtr4rd 2790 . . . . . . . . . . . . 13 (𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
160159adantl 482 . . . . . . . . . . . 12 (((𝑟𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
161 iffalse 4469 . . . . . . . . . . . . . 14 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
162 orc 864 . . . . . . . . . . . . . . . . 17 (𝑘𝑛 → (𝑘𝑛𝑘 = 𝑟))
163 orel2 888 . . . . . . . . . . . . . . . . 17 𝑘 = 𝑟 → ((𝑘𝑛𝑘 = 𝑟) → 𝑘𝑛))
164162, 163impbid2 225 . . . . . . . . . . . . . . . 16 𝑘 = 𝑟 → (𝑘𝑛 ↔ (𝑘𝑛𝑘 = 𝑟)))
165 elun 4084 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘𝑛𝑘 ∈ {𝑟}))
166 velsn 4578 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑟} ↔ 𝑘 = 𝑟)
167166orbi2i 910 . . . . . . . . . . . . . . . . 17 ((𝑘𝑛𝑘 ∈ {𝑟}) ↔ (𝑘𝑛𝑘 = 𝑟))
168165, 167bitr2i 275 . . . . . . . . . . . . . . . 16 ((𝑘𝑛𝑘 = 𝑟) ↔ 𝑘 ∈ (𝑛 ∪ {𝑟}))
169164, 168bitrdi 287 . . . . . . . . . . . . . . 15 𝑘 = 𝑟 → (𝑘𝑛𝑘 ∈ (𝑛 ∪ {𝑟})))
170169ifbid 4483 . . . . . . . . . . . . . 14 𝑘 = 𝑟 → if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
171161, 170eqtrd 2779 . . . . . . . . . . . . 13 𝑘 = 𝑟 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
172171adantl 482 . . . . . . . . . . . 12 (((𝑟𝐻 ∧ ¬ 𝑘 = 𝐻) ∧ ¬ 𝑘 = 𝑟) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
173160, 172pm2.61dan 810 . . . . . . . . . . 11 ((𝑟𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
174 iffalse 4469 . . . . . . . . . . . . 13 𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
175174ifeq2d 4480 . . . . . . . . . . . 12 𝑘 = 𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
176175adantl 482 . . . . . . . . . . 11 ((𝑟𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
177 iffalse 4469 . . . . . . . . . . . 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)))
178177adantl 482 . . . . . . . . . . 11 ((𝑟𝐻 ∧ ¬ 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)))
179173, 176, 1783eqtr4d 2789 . . . . . . . . . 10 ((𝑟𝐻 ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
180150, 179pm2.61dan 810 . . . . . . . . 9 (𝑟𝐻 → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))
181180mpoeq3dv 7363 . . . . . . . 8 (𝑟𝐻 → (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))
182181fveq2d 6787 . . . . . . 7 (𝑟𝐻 → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
18379, 182syl 17 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
18480, 139, 1833eqtr3d 2787 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
185184eqeq2d 2750 . . . 4 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) ↔ (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
186185biimpd 228 . . 3 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ((𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))))))
187 difss 4067 . . . 4 (𝑁 ∖ {𝐻}) ⊆ 𝑁
188 ssfi 8965 . . . 4 ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝐻}) ⊆ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
18947, 187, 188sylancl 586 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
1906, 12, 18, 24, 39, 186, 189findcard2d 8958 . 2 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))))
191 iba 528 . . . . . . . 8 (𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑙 = 𝐼𝑘 = 𝐻)))
192191ifbid 4483 . . . . . . 7 (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
193 iftrue 4466 . . . . . . 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 4466 . . . . . . . 8 ((𝑘 = 𝐻𝑙 = 𝐼) → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
195194orcs 872 . . . . . . 7 (𝑘 = 𝐻 → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
196192, 193, 1953eqtr4d 2789 . . . . . 6 (𝑘 = 𝐻 → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
197196adantl 482 . . . . 5 (((𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
198 iffalse 4469 . . . . . . 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)))
199198adantl 482 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 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 2952 . . . . . . . . . 10 𝑘 = 𝐻𝑘𝐻)
201200anim2i 617 . . . . . . . . 9 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
202201adantlr 712 . . . . . . . 8 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
203 eldifsn 4721 . . . . . . . 8 (𝑘 ∈ (𝑁 ∖ {𝐻}) ↔ (𝑘𝑁𝑘𝐻))
204202, 203sylibr 233 . . . . . . 7 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → 𝑘 ∈ (𝑁 ∖ {𝐻}))
205204iftrued 4468 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
206 biorf 934 . . . . . . . 8 𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑘 = 𝐻𝑙 = 𝐼)))
207 id 22 . . . . . . . . . . 11 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻)
208207intnand 489 . . . . . . . . . 10 𝑘 = 𝐻 → ¬ (𝑙 = 𝐼𝑘 = 𝐻))
209208iffalsed 4471 . . . . . . . . 9 𝑘 = 𝐻 → if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ) = 0 )
210209eqcomd 2745 . . . . . . . 8 𝑘 = 𝐻0 = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
211206, 210ifbieq1d 4484 . . . . . . 7 𝑘 = 𝐻 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
212211adantl 482 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
213199, 205, 2123eqtrd 2783 . . . . 5 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
214197, 213pm2.61dan 810 . . . 4 ((𝑘𝑁𝑙𝑁) → if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
215214mpoeq3ia 7362 . . 3 (𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))) = (𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
216215fveq2i 6786 . 2 (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙))))) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙))))
217190, 216eqtrdi 2795 1 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 844  w3a 1086   = wceq 1539  wcel 2107  wne 2944  Vcvv 3433  cdif 3885  cun 3886  wss 3888  c0 4257  ifcif 4460  {csn 4562   × cxp 5588  wf 6433  cfv 6437  (class class class)co 7284  cmpo 7286  m cmap 8624  Fincfn 8742  Basecbs 16921  +gcplusg 16971  .rcmulr 16972  0gc0g 17159  Mndcmnd 18394  1rcur 19746  Ringcrg 19792  CRingccrg 19793   Mat cmat 21563   maDet cmdat 21742   maAdju cmadu 21790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-addf 10959  ax-mulf 10960
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-of 7542  df-om 7722  df-1st 7840  df-2nd 7841  df-supp 7987  df-tpos 8051  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-2o 8307  df-er 8507  df-map 8626  df-pm 8627  df-ixp 8695  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-fsupp 9138  df-sup 9210  df-oi 9278  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-4 12047  df-5 12048  df-6 12049  df-7 12050  df-8 12051  df-9 12052  df-n0 12243  df-xnn0 12315  df-z 12329  df-dec 12447  df-uz 12592  df-rp 12740  df-fz 13249  df-fzo 13392  df-seq 13731  df-exp 13792  df-hash 14054  df-word 14227  df-lsw 14275  df-concat 14283  df-s1 14310  df-substr 14363  df-pfx 14393  df-splice 14472  df-reverse 14481  df-s2 14570  df-struct 16857  df-sets 16874  df-slot 16892  df-ndx 16904  df-base 16922  df-ress 16951  df-plusg 16984  df-mulr 16985  df-starv 16986  df-sca 16987  df-vsca 16988  df-ip 16989  df-tset 16990  df-ple 16991  df-ds 16993  df-unif 16994  df-hom 16995  df-cco 16996  df-0g 17161  df-gsum 17162  df-prds 17167  df-pws 17169  df-mre 17304  df-mrc 17305  df-acs 17307  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-mhm 18439  df-submnd 18440  df-efmnd 18517  df-grp 18589  df-minusg 18590  df-mulg 18710  df-subg 18761  df-ghm 18841  df-gim 18884  df-cntz 18932  df-oppg 18959  df-symg 18984  df-pmtr 19059  df-psgn 19108  df-evpm 19109  df-cmn 19397  df-abl 19398  df-mgp 19730  df-ur 19747  df-ring 19794  df-cring 19795  df-oppr 19871  df-dvdsr 19892  df-unit 19893  df-invr 19923  df-dvr 19934  df-rnghom 19968  df-drng 20002  df-subrg 20031  df-sra 20443  df-rgmod 20444  df-cnfld 20607  df-zring 20680  df-zrh 20714  df-dsmm 20948  df-frlm 20963  df-mat 21564  df-mdet 21743  df-madu 21792
This theorem is referenced by:  madutpos  21800
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