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Theorem maducoeval2 20265
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 2677 . . . . . . . 8 (𝑚 = ∅ → (𝑘𝑚𝑘 ∈ ∅))
21ifbid 4058 . . . . . . 7 (𝑚 = ∅ → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ ∅, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
32ifeq2d 4055 . . . . . 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))))
43mpt2eq3dv 6619 . . . . 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 6107 . . . 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 2620 . . 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 2677 . . . . . . . 8 (𝑚 = 𝑛 → (𝑘𝑚𝑘𝑛))
87ifbid 4058 . . . . . . 7 (𝑚 = 𝑛 → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
98ifeq2d 4055 . . . . . 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))))
109mpt2eq3dv 6619 . . . . 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 6107 . . . 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 2620 . . 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 2677 . . . . . . . 8 (𝑚 = (𝑛 ∪ {𝑟}) → (𝑘𝑚𝑘 ∈ (𝑛 ∪ {𝑟})))
1413ifbid 4058 . . . . . . 7 (𝑚 = (𝑛 ∪ {𝑟}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
1514ifeq2d 4055 . . . . . 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))))
1615mpt2eq3dv 6619 . . . . 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 6107 . . . 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 2620 . . 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 2677 . . . . . . . 8 (𝑚 = (𝑁 ∖ {𝐻}) → (𝑘𝑚𝑘 ∈ (𝑁 ∖ {𝐻})))
2019ifbid 4058 . . . . . . 7 (𝑚 = (𝑁 ∖ {𝐻}) → if(𝑘𝑚, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
2120ifeq2d 4055 . . . . . 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))))
2221mpt2eq3dv 6619 . . . . 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 6107 . . . 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 2620 . . 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 20264 . . . . 5 ((𝑀𝐵𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
32313adant1l 1310 . . . 4 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝐼(𝐽𝑀)𝐻) = (𝐷‘(𝑘𝑁, 𝑙𝑁 ↦ if(𝑘 = 𝐻, if(𝑙 = 𝐼, 1 , 0 ), (𝑘𝑀𝑙)))))
33 noel 3878 . . . . . . . 8 ¬ 𝑘 ∈ ∅
34 iffalse 4045 . . . . . . . 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 4055 . . . . . 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)))
3736mpt2eq3ia 6618 . . . . 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 6106 . . . 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, 38syl6eqr 2662 . . 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 2610 . . . . . . 7 (Base‘𝑅) = (Base‘𝑅)
41 eqid 2610 . . . . . . 7 (+g𝑅) = (+g𝑅)
42 eqid 2610 . . . . . . 7 (.r𝑅) = (.r𝑅)
43 simpl1l 1105 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑅 ∈ CRing)
44 simp1r 1079 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑀𝐵)
4525, 28matrcl 20037 . . . . . . . . . 10 (𝑀𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4645simpld 474 . . . . . . . . 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 1078 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → 𝑅 ∈ CRing)
5049ad2antrr 758 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑅 ∈ CRing)
51 crngring 18381 . . . . . . . . . 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 18392 . . . . . . . . 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 1106 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑀𝐵)
5625, 40, 28matbas2i 20047 . . . . . . . . . . 11 (𝑀𝐵𝑀 ∈ ((Base‘𝑅) ↑𝑚 (𝑁 × 𝑁)))
57 elmapi 7765 . . . . . . . . . . 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 3694 . . . . . . . . . . . 12 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6160ad2antll 761 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝑟 ∈ (𝑁 ∖ {𝐻}))
6261eldifad 3552 . . . . . . . . . 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 476 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → 𝑙𝑁)
6559, 63, 64fovrnd 6704 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → (𝑟𝑀𝑙) ∈ (Base‘𝑅))
6654, 65ifcld 4081 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)) ∈ (Base‘𝑅))
6740, 29ringidcl 18391 . . . . . . . . 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 4081 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → if(𝑙 = 𝐼, 1 , 0 ) ∈ (Base‘𝑅))
70543adant2 1073 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → 0 ∈ (Base‘𝑅))
7158fovrnda 6703 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ (𝑘𝑁𝑙𝑁)) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
72713impb 1252 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘𝑀𝑙) ∈ (Base‘𝑅))
7370, 72ifcld 4081 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) ∈ (Base‘𝑅))
7473, 72ifcld 4081 . . . . . . 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 1058 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐼𝑁)
7658, 62, 75fovrnd 6704 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → (𝑟𝑀𝐼) ∈ (Base‘𝑅))
77 simpl3 1059 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → 𝐻𝑁)
78 eldifsni 4261 . . . . . . . 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 20235 . . . . . 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 4083 . . . . . . . . . . . . . . . . 17 if(¬ 𝑙 = 𝐼, (𝑟𝑀𝑙), 0 ) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙))
8281eqcomi 2619 . . . . . . . . . . . . . . . 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 6636 . . . . . . . . . . . . . . . 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 18395 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝐼))
8752, 85, 86syl2anc 691 . . . . . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . 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 2647 . . . . . . . . . . . . . . . . . 18 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) ∧ 𝑙 = 𝐼) → ((𝑟𝑀𝐼)(.r𝑅) 1 ) = (𝑟𝑀𝑙))
9291ifeq1da 4066 . . . . . . . . . . . . . . . . 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 18411 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝑟𝑀𝐼) ∈ (Base‘𝑅)) → ((𝑟𝑀𝐼)(.r𝑅) 0 ) = 0 )
9452, 85, 93syl2anc 691 . . . . . . . . . . . . . . . . . 18 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑙𝑁) → ((𝑟𝑀𝐼)(.r𝑅) 0 ) = 0 )
9594ifeq2d 4055 . . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . . 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, 96syl5eq 2656 . . . . . . . . . . . . . . 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 6567 . . . . . . . . . . . . . 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 18379 . . . . . . . . . . . . . . . 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 437 . . . . . . . . . . . . . . . . 17 ((¬ 𝑙 = 𝐼 → ¬ 𝑙 = 𝐼) ↔ ¬ (¬ 𝑙 = 𝐼𝑙 = 𝐼))
103101, 102mpbi 219 . . . . . . . . . . . . . . . 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 20261 . . . . . . . . . . . . . . 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 1318 . . . . . . . . . . . . . 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 pm2.1 432 . . . . . . . . . . . . . . 15 𝑙 = 𝐼𝑙 = 𝐼)
108 iftrue 4042 . . . . . . . . . . . . . . 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 2650 . . . . . . . . . . . . 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 1073 . . . . . . . . . . . 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 6556 . . . . . . . . . . . . 13 (𝑘 = 𝑟 → (𝑘𝑀𝑙) = (𝑟𝑀𝑙))
113112eqeq2d 2620 . . . . . . . . . . . 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 236 . . . . . . . . . . 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 444 . . . . . . . . . 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 4042 . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟 = 𝐻)
1191183ad2ant1 1075 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → ¬ 𝑟 = 𝐻)
120 eqeq1 2614 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘 = 𝐻𝑟 = 𝐻))
121120notbid 307 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘 = 𝐻 ↔ ¬ 𝑟 = 𝐻))
122119, 121syl5ibrcom 236 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘 = 𝐻))
123122imp 444 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘 = 𝐻)
124123iffalsed 4047 . . . . . . . . . . 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 3695 . . . . . . . . . . . . . . . 16 (𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛) → ¬ 𝑟𝑛)
126125ad2antll 761 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) → ¬ 𝑟𝑛)
1271263ad2ant1 1075 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → ¬ 𝑟𝑛)
128 eleq1 2676 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟 → (𝑘𝑛𝑟𝑛))
129128notbid 307 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → (¬ 𝑘𝑛 ↔ ¬ 𝑟𝑛))
130127, 129syl5ibrcom 236 . . . . . . . . . . . . 13 (((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) → (𝑘 = 𝑟 → ¬ 𝑘𝑛))
131130imp 444 . . . . . . . . . . . 12 ((((((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) ∧ (𝑛 ⊆ (𝑁 ∖ {𝐻}) ∧ 𝑟 ∈ ((𝑁 ∖ {𝐻}) ∖ 𝑛))) ∧ 𝑘𝑁𝑙𝑁) ∧ 𝑘 = 𝑟) → ¬ 𝑘𝑛)
132131iffalsed 4047 . . . . . . . . . . 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 2644 . . . . . . . . . 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 2654 . . . . . . . . 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 4045 . . . . . . . . . 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 828 . . . . . . . 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))))
138137mpt2eq3dva 6617 . . . . . . 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 6107 . . . . . 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 2845 . . . . . . . . . . . . . . 15 (𝑘 = 𝐻 → (𝑟𝑘𝑟𝐻))
141140biimparc 503 . . . . . . . . . . . . . 14 ((𝑟𝐻𝑘 = 𝐻) → 𝑟𝑘)
142141necomd 2837 . . . . . . . . . . . . 13 ((𝑟𝐻𝑘 = 𝐻) → 𝑘𝑟)
143142neneqd 2787 . . . . . . . . . . . 12 ((𝑟𝐻𝑘 = 𝐻) → ¬ 𝑘 = 𝑟)
144143iffalsed 4047 . . . . . . . . . . 11 ((𝑟𝐻𝑘 = 𝐻) → if(𝑘 = 𝑟, if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)), if(𝑙 = 𝐼, 1 , 0 )) = if(𝑙 = 𝐼, 1 , 0 ))
145 iftrue 4042 . . . . . . . . . . . . 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 4055 . . . . . . . . . . 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 4042 . . . . . . . . . . . 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 2654 . . . . . . . . . 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 4055 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑟𝑀𝑙)))
152 vsnid 4156 . . . . . . . . . . . . . . . . 17 𝑟 ∈ {𝑟}
153 elun2 3743 . . . . . . . . . . . . . . . . 17 (𝑟 ∈ {𝑟} → 𝑟 ∈ (𝑛 ∪ {𝑟}))
154152, 153ax-mp 5 . . . . . . . . . . . . . . . 16 𝑟 ∈ (𝑛 ∪ {𝑟})
155 eleq1 2676 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑟 → (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ 𝑟 ∈ (𝑛 ∪ {𝑟})))
156154, 155mpbiri 247 . . . . . . . . . . . . . . 15 (𝑘 = 𝑟𝑘 ∈ (𝑛 ∪ {𝑟}))
157156iftrued 4044 . . . . . . . . . . . . . 14 (𝑘 = 𝑟 → if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
158 iftrue 4042 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 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 4045 . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . . 17 (𝑘𝑛 → (𝑘𝑛𝑘 = 𝑟))
163 orel2 397 . . . . . . . . . . . . . . . . 17 𝑘 = 𝑟 → ((𝑘𝑛𝑘 = 𝑟) → 𝑘𝑛))
164162, 163impbid2 215 . . . . . . . . . . . . . . . 16 𝑘 = 𝑟 → (𝑘𝑛 ↔ (𝑘𝑛𝑘 = 𝑟)))
165 elun 3715 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑛 ∪ {𝑟}) ↔ (𝑘𝑛𝑘 ∈ {𝑟}))
166 velsn 4141 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ {𝑟} ↔ 𝑘 = 𝑟)
167166orbi2i 540 . . . . . . . . . . . . . . . . 17 ((𝑘𝑛𝑘 ∈ {𝑟}) ↔ (𝑘𝑛𝑘 = 𝑟))
168165, 167bitr2i 264 . . . . . . . . . . . . . . . 16 ((𝑘𝑛𝑘 = 𝑟) ↔ 𝑘 ∈ (𝑛 ∪ {𝑟}))
169164, 168syl6bb 275 . . . . . . . . . . . . . . 15 𝑘 = 𝑟 → (𝑘𝑛𝑘 ∈ (𝑛 ∪ {𝑟})))
170169ifbid 4058 . . . . . . . . . . . . . 14 𝑘 = 𝑟 → if(𝑘𝑛, if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑘 ∈ (𝑛 ∪ {𝑟}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)))
171161, 170eqtrd 2644 . . . . . . . . . . . . 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 828 . . . . . . . . . . 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 4045 . . . . . . . . . . . . 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 4055 . . . . . . . . . . . 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 4045 . . . . . . . . . . . 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 2654 . . . . . . . . . 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 828 . . . . . . . . 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))))
181180mpt2eq3dv 6619 . . . . . . . 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 6107 . . . . . . 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 2652 . . . . 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 2620 . . . 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 218 . . 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 3699 . . . 4 (𝑁 ∖ {𝐻}) ⊆ 𝑁
188 ssfi 8065 . . . 4 ((𝑁 ∈ Fin ∧ (𝑁 ∖ {𝐻}) ⊆ 𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
18947, 187, 188sylancl 693 . . 3 (((𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝐼𝑁𝐻𝑁) → (𝑁 ∖ {𝐻}) ∈ Fin)
1906, 12, 18, 24, 39, 186, 189findcard2d 8087 . 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 523 . . . . . . . 8 (𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑙 = 𝐼𝑘 = 𝐻)))
192191ifbid 4058 . . . . . . 7 (𝑘 = 𝐻 → if(𝑙 = 𝐼, 1 , 0 ) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
193 iftrue 4042 . . . . . . 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 4042 . . . . . . . 8 ((𝑘 = 𝐻𝑙 = 𝐼) → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
195194orcs 408 . . . . . . 7 (𝑘 = 𝐻 → if((𝑘 = 𝐻𝑙 = 𝐼), if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ), (𝑘𝑀𝑙)) = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
196192, 193, 1953eqtr4d 2654 . . . . . 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 4045 . . . . . . 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 id 22 . . . . . . . . . . 11 𝑘 = 𝐻 → ¬ 𝑘 = 𝐻)
201200neqned 2789 . . . . . . . . . 10 𝑘 = 𝐻𝑘𝐻)
202201anim2i 591 . . . . . . . . 9 ((𝑘𝑁 ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
203202adantlr 747 . . . . . . . 8 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → (𝑘𝑁𝑘𝐻))
204 eldifsn 4260 . . . . . . . 8 (𝑘 ∈ (𝑁 ∖ {𝐻}) ↔ (𝑘𝑁𝑘𝐻))
205203, 204sylibr 223 . . . . . . 7 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → 𝑘 ∈ (𝑁 ∖ {𝐻}))
206205iftrued 4044 . . . . . 6 (((𝑘𝑁𝑙𝑁) ∧ ¬ 𝑘 = 𝐻) → if(𝑘 ∈ (𝑁 ∖ {𝐻}), if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)), (𝑘𝑀𝑙)) = if(𝑙 = 𝐼, 0 , (𝑘𝑀𝑙)))
207 biorf 419 . . . . . . . 8 𝑘 = 𝐻 → (𝑙 = 𝐼 ↔ (𝑘 = 𝐻𝑙 = 𝐼)))
208200intnand 953 . . . . . . . . . 10 𝑘 = 𝐻 → ¬ (𝑙 = 𝐼𝑘 = 𝐻))
209208iffalsed 4047 . . . . . . . . 9 𝑘 = 𝐻 → if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ) = 0 )
210209eqcomd 2616 . . . . . . . 8 𝑘 = 𝐻0 = if((𝑙 = 𝐼𝑘 = 𝐻), 1 , 0 ))
211207, 210ifbieq1d 4059 . . . . . . 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, 206, 2123eqtrd 2648 . . . . 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 828 . . . 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)))
215214mpt2eq3ia 6618 . . 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 6106 . 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, 216syl6eq 2660 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  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  Vcvv 3173  cdif 3537  cun 3538  wss 3540  c0 3874  ifcif 4036  {csn 4125   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  cmpt2 6551  𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Mndcmnd 17117  1rcur 18324  Ringcrg 18370  CRingccrg 18371   Mat cmat 20032   maDet cmdat 20209   maAdju cmadu 20257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-addf 9894  ax-mulf 9895
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-subg 17414  df-ghm 17481  df-gim 17524  df-cntz 17573  df-oppg 17599  df-symg 17621  df-pmtr 17685  df-psgn 17734  df-evpm 17735  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-sra 18993  df-rgmod 18994  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-dsmm 19895  df-frlm 19910  df-mat 20033  df-mdet 20210  df-madu 20259
This theorem is referenced by:  madutpos  20267
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