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Theorem insubm 12826
Description: The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)
Assertion
Ref Expression
insubm ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀))

Proof of Theorem insubm
Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 12816 . . 3 (𝐴 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2 ssinss1 3364 . . . . . . . . 9 (𝐴 ⊆ (Base‘𝑀) → (𝐴𝐵) ⊆ (Base‘𝑀))
323ad2ant1 1018 . . . . . . . 8 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → (𝐴𝐵) ⊆ (Base‘𝑀))
43ad2antrl 490 . . . . . . 7 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → (𝐴𝐵) ⊆ (Base‘𝑀))
5 elin 3318 . . . . . . . . . . . . 13 ((0g𝑀) ∈ (𝐴𝐵) ↔ ((0g𝑀) ∈ 𝐴 ∧ (0g𝑀) ∈ 𝐵))
65simplbi2com 1444 . . . . . . . . . . . 12 ((0g𝑀) ∈ 𝐵 → ((0g𝑀) ∈ 𝐴 → (0g𝑀) ∈ (𝐴𝐵)))
763ad2ant2 1019 . . . . . . . . . . 11 ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → ((0g𝑀) ∈ 𝐴 → (0g𝑀) ∈ (𝐴𝐵)))
87com12 30 . . . . . . . . . 10 ((0g𝑀) ∈ 𝐴 → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (0g𝑀) ∈ (𝐴𝐵)))
983ad2ant2 1019 . . . . . . . . 9 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (0g𝑀) ∈ (𝐴𝐵)))
109imp 124 . . . . . . . 8 (((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)) → (0g𝑀) ∈ (𝐴𝐵))
1110adantl 277 . . . . . . 7 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → (0g𝑀) ∈ (𝐴𝐵))
12 elin 3318 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
13 elin 3318 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
1412, 13anbi12i 460 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)))
15 oveq1 5881 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(+g𝑀)𝑏) = (𝑥(+g𝑀)𝑏))
1615eleq1d 2246 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(+g𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g𝑀)𝑏) ∈ 𝐴))
17 oveq2 5882 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → (𝑥(+g𝑀)𝑏) = (𝑥(+g𝑀)𝑦))
1817eleq1d 2246 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑦 → ((𝑥(+g𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝐴))
19 simpl 109 . . . . . . . . . . . . . . . . 17 ((𝑥𝐴𝑥𝐵) → 𝑥𝐴)
2019adantr 276 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑥𝐴)
21 eqidd 2178 . . . . . . . . . . . . . . . 16 ((((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) ∧ 𝑎 = 𝑥) → 𝐴 = 𝐴)
22 simpl 109 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝑦𝐵) → 𝑦𝐴)
2322adantl 277 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑦𝐴)
2416, 18, 20, 21, 23rspc2vd 3125 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴 → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
2524com12 30 . . . . . . . . . . . . . 14 (∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴 → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
26253ad2ant3 1020 . . . . . . . . . . . . 13 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
2726ad2antrl 490 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
2827imp 124 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵))) → (𝑥(+g𝑀)𝑦) ∈ 𝐴)
2915eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → ((𝑎(+g𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g𝑀)𝑏) ∈ 𝐵))
3017eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → ((𝑥(+g𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝐵))
31 simpr 110 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
3231adantr 276 . . . . . . . . . . . . . . . . 17 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑥𝐵)
33 eqidd 2178 . . . . . . . . . . . . . . . . 17 ((((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) ∧ 𝑎 = 𝑥) → 𝐵 = 𝐵)
34 simpr 110 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐴𝑦𝐵) → 𝑦𝐵)
3534adantl 277 . . . . . . . . . . . . . . . . 17 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑦𝐵)
3629, 30, 32, 33, 35rspc2vd 3125 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵 → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
3736com12 30 . . . . . . . . . . . . . . 15 (∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵 → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
38373ad2ant3 1020 . . . . . . . . . . . . . 14 ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
3938adantl 277 . . . . . . . . . . . . 13 (((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
4039adantl 277 . . . . . . . . . . . 12 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
4140imp 124 . . . . . . . . . . 11 (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵))) → (𝑥(+g𝑀)𝑦) ∈ 𝐵)
4228, 41elind 3320 . . . . . . . . . 10 (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵))) → (𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵))
4342ex 115 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵)))
4414, 43biimtrid 152 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 ∈ (𝐴𝐵)) → (𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵)))
4544ralrimivv 2558 . . . . . . 7 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵))
464, 11, 453jca 1177 . . . . . 6 ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))) → ((𝐴𝐵) ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵)))
4746ex 115 . . . . 5 (𝑀 ∈ Mnd → (((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)) → ((𝐴𝐵) ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵))))
48 eqid 2177 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
49 eqid 2177 . . . . . . 7 (0g𝑀) = (0g𝑀)
50 eqid 2177 . . . . . . 7 (+g𝑀) = (+g𝑀)
5148, 49, 50issubm 12817 . . . . . 6 (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴)))
5248, 49, 50issubm 12817 . . . . . 6 (𝑀 ∈ Mnd → (𝐵 ∈ (SubMnd‘𝑀) ↔ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)))
5351, 52anbi12d 473 . . . . 5 (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) ↔ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))))
5448, 49, 50issubm 12817 . . . . 5 (𝑀 ∈ Mnd → ((𝐴𝐵) ∈ (SubMnd‘𝑀) ↔ ((𝐴𝐵) ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵))))
5547, 53, 543imtr4d 203 . . . 4 (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀)))
5655expd 258 . . 3 (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴𝐵) ∈ (SubMnd‘𝑀))))
571, 56mpcom 36 . 2 (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴𝐵) ∈ (SubMnd‘𝑀)))
5857imp 124 1 ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978  wcel 2148  wral 2455  cin 3128  wss 3129  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  0gc0g 12695  Mndcmnd 12771  SubMndcsubmnd 12804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-inn 8918  df-ndx 12459  df-slot 12460  df-base 12462  df-submnd 12806
This theorem is referenced by: (None)
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