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Theorem maducoeval2 22661
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 2827 . . . . . . . 8 (𝑚 = ∅ → (𝑘𝑚𝑘 ∈ ∅))
21ifbid 4553 . . . . . . 7 (𝑚 = ∅ → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
32ifeq2d 4550 . . . . . 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 7511 . . . . 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 6910 . . . 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 2745 . . 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 2827 . . . . . . . 8 (𝑚 = 𝑛 → (𝑘𝑚𝑘𝑛))
87ifbid 4553 . . . . . . 7 (𝑚 = 𝑛 → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
98ifeq2d 4550 . . . . . 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 7511 . . . . 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 6910 . . . 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 2745 . . 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 2827 . . . . . . . 8 (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘𝑚𝑘 ∈ (𝑛 ∪ {𝑟})))
1413ifbid 4553 . . . . . . 7 (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
1514ifeq2d 4550 . . . . . 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 7511 . . . . 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 6910 . . . 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 2745 . . 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 2827 . . . . . . . 8 (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘𝑚𝑘 ∈ (𝑁 ∖ {𝐻})))
2019ifbid 4553 . . . . . . 7 (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
2120ifeq2d 4550 . . . . . 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 7511 . . . . 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 6910 . . . 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 2745 . . 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 22660 . . . . 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 4343 . . . . . . . 8 ¬ 𝑘 ∈ ∅
34 iffalse 4539 . . . . . . . 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 4550 . . . . . 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 7510 . . . . 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 6909 . . . 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 2792 . . 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 2734 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
41 eqid 2734 . . . . . . 7 (+g𝑅) = (+g𝑅)
42 eqid 2734 . . . . . . 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 22431 . . . . . . . . . 10 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4645simpld 494 . . . . . . . . 9 (𝑀𝐵𝑁 ∈ Fin)
4744, 46syl 17 . . . . . . . 8 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑁 ∈ Fin)
4847adantr 480 . . . . . . 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 726 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑅 ∈ CRing)
51 crngring 20262 . . . . . . . . . 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 20280 . . . . . . . . 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 22443 . . . . . . . . . . 11 (𝑀𝐵𝑀 ∈ ((Base‘𝑅) ↑m (𝑁 × 𝑁)))
57 elmapi 8887 . . . . . . . . . . 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 480 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑀:(𝑁 × 𝑁)⟶(Base‘𝑅))
60 eldifi 4140 . . . . . . . . . . . 12 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6160ad2antll 729 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6261eldifad 3974 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟𝑁)
6362adantr 480 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑟𝑁)
64 simpr 484 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑙𝑁)
6559, 63, 64fovcdmd 7604 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (𝑟𝑀𝑙) ∈ (Base‘𝑅))
6654, 65ifcld 4576 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) ∈ (Base‘𝑅))
6740, 29ringidcl 20279 . . . . . . . . 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 4576 . . . . . . 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))
7158fovcdmda 7603 . . . . . . . . . 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 4576 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
7473, 72ifcld 4576 . . . . . . 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, 75fovcdmd 7604 . . . . . . 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 4794 . . . . . . . 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 22631 . . . . . 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 4582 . . . . . . . . . . . . . . . . 17 if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))
8281eqcomi 2743 . . . . . . . . . . . . . . . 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 7531 . . . . . . . . . . . . . . . 16 ((𝑟𝑀𝐼)(.r𝑅)if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, ((𝑟𝑀𝐼)(.r𝑅) 1 ), ((𝑟𝑀𝐼)(.r𝑅) 0 ))
8576adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
8640, 42, 29ringridm 20283 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝐼))
89 oveq2 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑙 = 𝐼 → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
9089adantl 481 . . . . . . . . . . . . . . . . . . 19 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → (𝑟𝑀𝑙) = (𝑟𝑀𝐼))
9188, 90eqtr4d 2777 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝑙))
9291ifeq1da 4561 . . . . . . . . . . . . . . . . 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 20307 . . . . . . . . . . . . . . . . . . 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 4550 . . . . . . . . . . . . . . . . 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 2774 . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . 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 20260 . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . 17 ((¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼) ↔ ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼))
103101, 102mpbi 230 . . . . . . . . . . . . . . . 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 22657 . . . . . . . . . . . . . . 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 pm2.1 896 . . . . . . . . . . . . . . 15 𝑙 = 𝐼𝑙 = 𝐼)
108 iftrue 4536 . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . 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 7437 . . . . . . . . . . . . 13 (𝑘 = 𝑟 → (𝑘𝑀𝑙) = (𝑟𝑀𝑙))
113112eqeq2d 2745 . . . . . . . . . . . 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 247 . . . . . . . . . . 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 406 . . . . . . . . . 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 4536 . . . . . . . . . . 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 481 . . . . . . . . . 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 2942 . . . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘 = 𝐻𝑟 = 𝐻))
121120notbid 318 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻))
122119, 121syl5ibrcom 247 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 = 𝐻))
123122imp 406 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 = 𝐻)
124123iffalsed 4541 . . . . . . . . . . 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 4141 . . . . . . . . . . . . . . . 16 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟𝑛)
126125ad2antll 729 . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘𝑛𝑟𝑛))
129128notbid 318 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘𝑛 ↔ ¬ 𝑟𝑛))
130127, 129syl5ibrcom 247 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘𝑛))
131130imp 406 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘𝑛)
132131iffalsed 4541 . . . . . . . . . . 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 2774 . . . . . . . . . 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 2784 . . . . . . . . 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 4539 . . . . . . . . . 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 481 . . . . . . . . 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 813 . . . . . . . 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 7509 . . . . . . 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 6910 . . . . . 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 3001 . . . . . . . . . . . . . . 15 (𝑘 = 𝐻 → (𝑟𝑘𝑟𝐻))
141140biimparc 479 . . . . . . . . . . . . . 14 ((𝑟𝐻𝑘 = 𝐻) → 𝑟𝑘)
142141necomd 2993 . . . . . . . . . . . . 13 ((𝑟𝐻𝑘 = 𝐻) → 𝑘𝑟)
143142neneqd 2942 . . . . . . . . . . . 12 ((𝑟𝐻𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144143iffalsed 4541 . . . . . . . . . . 11 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, 1 , 0 ))
145 iftrue 4536 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 4550 . . . . . . . . . . 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 4536 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 2784 . . . . . . . . . 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 4550 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152 vsnid 4667 . . . . . . . . . . . . . . . . 17 𝑟 ∈ {𝑟}
153 elun2 4192 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16 𝑟 ∈ (𝑛 ∪ {𝑟})
155 eleq1w 2821 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156154, 155mpbiri 258 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟𝑘 ∈ (𝑛 ∪ {𝑟}))
157156iftrued 4538 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158 iftrue 4536 . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 4539 . . . . . . . . . . . . . 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 867 . . . . . . . . . . . . . . . . 17 (𝑘𝑛 → (𝑘𝑛𝑘 = 𝑟))
163 orel2 890 . . . . . . . . . . . . . . . . 17 𝑘 = 𝑟 → ((𝑘𝑛𝑘 = 𝑟) → 𝑘𝑛))
164162, 163impbid2 226 . . . . . . . . . . . . . . . 16 𝑘 = 𝑟 → (𝑘𝑛 ↔ (𝑘𝑛𝑘 = 𝑟)))
165 elun 4162 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘𝑛𝑘 ∈ {𝑟}))
166 velsn 4646 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑟} ↔ 𝑘 = 𝑟)
167166orbi2i 912 . . . . . . . . . . . . . . . . 17 ((𝑘𝑛𝑘 ∈ {𝑟}) ↔ (𝑘𝑛𝑘 = 𝑟))
168165, 167bitr2i 276 . . . . . . . . . . . . . . . 16 ((𝑘𝑛𝑘 = 𝑟) ↔ 𝑘 ∈ (𝑛 ∪ {𝑟}))
169164, 168bitrdi 287 . . . . . . . . . . . . . . 15 𝑘 = 𝑟 → (𝑘𝑛𝑘 ∈ (𝑛 ∪ {𝑟})))
170169ifbid 4553 . . . . . . . . . . . . . 14 𝑘 = 𝑟 → if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
171161, 170eqtrd 2774 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 813 . . . . . . . . . . 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 4539 . . . . . . . . . . . . 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 4550 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 4539 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 2784 . . . . . . . . . 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 813 . . . . . . . . 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 7511 . . . . . . . 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 6910 . . . . . . 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 2782 . . . . 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 2745 . . . 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 229 . . 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 4145 . . . 4 (𝑁 ∖ {𝐻}) ⊆ 𝑁
188 ssfi 9211 . . . 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 9204 . 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 527 . . . . . . . 8 (𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑙 = 𝐼𝑘 = 𝐻)))
192191ifbid 4553 . . . . . . 7 (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
193 iftrue 4536 . . . . . . 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 4536 . . . . . . . 8 ((𝑘 = 𝐻𝑙 = 𝐼) → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
195194orcs 875 . . . . . . 7 (𝑘 = 𝐻 → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
196192, 193, 1953eqtr4d 2784 . . . . . 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 481 . . . . 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 4539 . . . . . . 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 481 . . . . . 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 2945 . . . . . . . . . 10 𝑘 = 𝐻𝑘𝐻)
201200anim2i 617 . . . . . . . . 9 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
202201adantlr 715 . . . . . . . 8 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
203 eldifsn 4790 . . . . . . . 8 (𝑘 ∈ (𝑁 ∖ {𝐻}) ↔ (𝑘𝑁𝑘𝐻))
204202, 203sylibr 234 . . . . . . 7 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → 𝑘 ∈ (𝑁 ∖ {𝐻}))
205204iftrued 4538 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
206 biorf 936 . . . . . . . 8 𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑘 = 𝐻𝑙 = 𝐼)))
207 id 22 . . . . . . . . . . 11 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻)
208207intnand 488 . . . . . . . . . 10 𝑘 = 𝐻 → ¬ (𝑙 = 𝐼𝑘 = 𝐻))
209208iffalsed 4541 . . . . . . . . 9 𝑘 = 𝐻 → if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ) = 0 )
210209eqcomd 2740 . . . . . . . 8 𝑘 = 𝐻0 = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
211206, 210ifbieq1d 4554 . . . . . . 7 𝑘 = 𝐻 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
212211adantl 481 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)))
213199, 205, 2123eqtrd 2778 . . . . 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 813 . . . 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 7510 . . 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 6909 . 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 2790 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 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wne 2937  Vcvv 3477  cdif 3959  cun 3960  wss 3962  c0 4338  ifcif 4530  {csn 4630   × cxp 5686  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432  m cmap 8864  Fincfn 8983  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  0gc0g 17485  Mndcmnd 18759  1rcur 20198  Ringcrg 20250  CRingccrg 20251   Mat cmat 22426   maDet cmdat 22605   maAdju cmadu 22653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1508  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-word 14549  df-lsw 14597  df-concat 14605  df-s1 14630  df-substr 14675  df-pfx 14705  df-splice 14784  df-reverse 14793  df-s2 14883  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-efmnd 18894  df-grp 18966  df-minusg 18967  df-mulg 19098  df-subg 19153  df-ghm 19243  df-gim 19289  df-cntz 19347  df-oppg 19376  df-symg 19401  df-pmtr 19474  df-psgn 19523  df-evpm 19524  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-drng 20747  df-sra 21189  df-rgmod 21190  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-dsmm 21769  df-frlm 21784  df-mat 22427  df-mdet 22606  df-madu 22655
This theorem is referenced by:  madutpos  22663
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