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Theorem insubm 18099
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 18083 . . 3 (𝐴 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2 ssinss1 4128 . . . . . . . . 9 (𝐴 ⊆ (Base‘𝑀) → (𝐴𝐵) ⊆ (Base‘𝑀))
323ad2ant1 1134 . . . . . . . 8 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → (𝐴𝐵) ⊆ (Base‘𝑀))
43ad2antrl 728 . . . . . . 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 3859 . . . . . . . . . . . . 13 ((0g𝑀) ∈ (𝐴𝐵) ↔ ((0g𝑀) ∈ 𝐴 ∧ (0g𝑀) ∈ 𝐵))
65simplbi2com 506 . . . . . . . . . . . 12 ((0g𝑀) ∈ 𝐵 → ((0g𝑀) ∈ 𝐴 → (0g𝑀) ∈ (𝐴𝐵)))
763ad2ant2 1135 . . . . . . . . . . 11 ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → ((0g𝑀) ∈ 𝐴 → (0g𝑀) ∈ (𝐴𝐵)))
87com12 32 . . . . . . . . . 10 ((0g𝑀) ∈ 𝐴 → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (0g𝑀) ∈ (𝐴𝐵)))
983ad2ant2 1135 . . . . . . . . 9 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (0g𝑀) ∈ (𝐴𝐵)))
109imp 410 . . . . . . . 8 (((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)) → (0g𝑀) ∈ (𝐴𝐵))
1110adantl 485 . . . . . . 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 3859 . . . . . . . . . 10 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
13 elin 3859 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
1412, 13anbi12i 630 . . . . . . . . 9 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)))
15 oveq1 7177 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → (𝑎(+g𝑀)𝑏) = (𝑥(+g𝑀)𝑏))
1615eleq1d 2817 . . . . . . . . . . . . . . . 16 (𝑎 = 𝑥 → ((𝑎(+g𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g𝑀)𝑏) ∈ 𝐴))
17 oveq2 7178 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → (𝑥(+g𝑀)𝑏) = (𝑥(+g𝑀)𝑦))
1817eleq1d 2817 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑦 → ((𝑥(+g𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝐴))
19 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝑥𝐴𝑥𝐵) → 𝑥𝐴)
2019adantr 484 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑥𝐴)
21 eqidd 2739 . . . . . . . . . . . . . . . 16 ((((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) ∧ 𝑎 = 𝑥) → 𝐴 = 𝐴)
22 simpl 486 . . . . . . . . . . . . . . . . 17 ((𝑦𝐴𝑦𝐵) → 𝑦𝐴)
2322adantl 485 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑦𝐴)
2416, 18, 20, 21, 23rspc2vd 3839 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴 → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
2524com12 32 . . . . . . . . . . . . . 14 (∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴 → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
26253ad2ant3 1136 . . . . . . . . . . . . 13 ((𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐴))
2726ad2antrl 728 . . . . . . . . . . . 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 410 . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑥 → ((𝑎(+g𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g𝑀)𝑏) ∈ 𝐵))
3017eleq1d 2817 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑦 → ((𝑥(+g𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝐵))
31 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑥𝐴𝑥𝐵) → 𝑥𝐵)
3231adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑥𝐵)
33 eqidd 2739 . . . . . . . . . . . . . . . . 17 ((((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) ∧ 𝑎 = 𝑥) → 𝐵 = 𝐵)
34 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝑦𝐴𝑦𝐵) → 𝑦𝐵)
3534adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → 𝑦𝐵)
3629, 30, 32, 33, 35rspc2vd 3839 . . . . . . . . . . . . . . . 16 (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵 → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
3736com12 32 . . . . . . . . . . . . . . 15 (∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵 → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
38373ad2ant3 1136 . . . . . . . . . . . . . 14 ((𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵) → (((𝑥𝐴𝑥𝐵) ∧ (𝑦𝐴𝑦𝐵)) → (𝑥(+g𝑀)𝑦) ∈ 𝐵))
3938adantl 485 . . . . . . . . . . . . 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 485 . . . . . . . . . . . 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 410 . . . . . . . . . . 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 4084 . . . . . . . . . 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 416 . . . . . . . . 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, 43syl5bi 245 . . . . . . . 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 3102 . . . . . . 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 1129 . . . . . 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 416 . . . . 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 2738 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
49 eqid 2738 . . . . . . 7 (0g𝑀) = (0g𝑀)
50 eqid 2738 . . . . . . 7 (+g𝑀) = (+g𝑀)
5148, 49, 50issubm 18084 . . . . . 6 (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐴 ∧ ∀𝑎𝐴𝑏𝐴 (𝑎(+g𝑀)𝑏) ∈ 𝐴)))
5248, 49, 50issubm 18084 . . . . . 6 (𝑀 ∈ Mnd → (𝐵 ∈ (SubMnd‘𝑀) ↔ (𝐵 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝐵 ∧ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)))
5351, 52anbi12d 634 . . . . 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 18084 . . . . 5 (𝑀 ∈ Mnd → ((𝐴𝐵) ∈ (SubMnd‘𝑀) ↔ ((𝐴𝐵) ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)∀𝑦 ∈ (𝐴𝐵)(𝑥(+g𝑀)𝑦) ∈ (𝐴𝐵))))
5547, 53, 543imtr4d 297 . . . 4 (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀)))
5655expd 419 . . 3 (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴𝐵) ∈ (SubMnd‘𝑀))))
571, 56mpcom 38 . 2 (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴𝐵) ∈ (SubMnd‘𝑀)))
5857imp 410 1 ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴𝐵) ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088  wcel 2114  wral 3053  cin 3842  wss 3843  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  0gc0g 16816  Mndcmnd 18027  SubMndcsubmnd 18071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-submnd 18073
This theorem is referenced by:  symgsubmefmnd  18644
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