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Theorem ispridl2 37419
Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 37451 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ispridl2.1 𝐺 = (1st β€˜π‘…)
ispridl2.2 𝐻 = (2nd β€˜π‘…)
ispridl2.3 𝑋 = ran 𝐺
Assertion
Ref Expression
ispridl2 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) β†’ 𝑃 ∈ (PrIdlβ€˜π‘…))
Distinct variable groups:   𝑅,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑋,π‘Ž,𝑏
Allowed substitution hints:   𝐺(π‘Ž,𝑏)   𝐻(π‘Ž,𝑏)

Proof of Theorem ispridl2
Dummy variables π‘Ÿ 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ispridl2.1 . . . . . . . . . . . . . 14 𝐺 = (1st β€˜π‘…)
2 ispridl2.3 . . . . . . . . . . . . . 14 𝑋 = ran 𝐺
31, 2idlss 37397 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ π‘Ÿ ∈ (Idlβ€˜π‘…)) β†’ π‘Ÿ βŠ† 𝑋)
4 ssralv 4045 . . . . . . . . . . . . 13 (π‘Ÿ βŠ† 𝑋 β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
53, 4syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ π‘Ÿ ∈ (Idlβ€˜π‘…)) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
65adantrr 714 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (π‘Ÿ ∈ (Idlβ€˜π‘…) ∧ 𝑠 ∈ (Idlβ€˜π‘…))) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
71, 2idlss 37397 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idlβ€˜π‘…)) β†’ 𝑠 βŠ† 𝑋)
8 ssralv 4045 . . . . . . . . . . . . . 14 (𝑠 βŠ† 𝑋 β†’ (βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
98ralimdv 3163 . . . . . . . . . . . . 13 (𝑠 βŠ† 𝑋 β†’ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
107, 9syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idlβ€˜π‘…)) β†’ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
1110adantrl 713 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ (π‘Ÿ ∈ (Idlβ€˜π‘…) ∧ 𝑠 ∈ (Idlβ€˜π‘…))) β†’ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
126, 11syld 47 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ (π‘Ÿ ∈ (Idlβ€˜π‘…) ∧ 𝑠 ∈ (Idlβ€˜π‘…))) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
1312adantlr 712 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idlβ€˜π‘…)) ∧ (π‘Ÿ ∈ (Idlβ€˜π‘…) ∧ 𝑠 ∈ (Idlβ€˜π‘…))) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
14 r19.26-2 3132 . . . . . . . . . . 11 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 ∧ ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) ↔ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 ∧ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
15 pm3.35 800 . . . . . . . . . . . . 13 (((π‘Žπ»π‘) ∈ 𝑃 ∧ ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
16152ralimi 3117 . . . . . . . . . . . 12 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 ∧ ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
17 2ralor 3222 . . . . . . . . . . . . 13 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (βˆ€π‘Ž ∈ π‘Ÿ π‘Ž ∈ 𝑃 ∨ βˆ€π‘ ∈ 𝑠 𝑏 ∈ 𝑃))
18 dfss3 3965 . . . . . . . . . . . . . 14 (π‘Ÿ βŠ† 𝑃 ↔ βˆ€π‘Ž ∈ π‘Ÿ π‘Ž ∈ 𝑃)
19 dfss3 3965 . . . . . . . . . . . . . 14 (𝑠 βŠ† 𝑃 ↔ βˆ€π‘ ∈ 𝑠 𝑏 ∈ 𝑃)
2018, 19orbi12i 911 . . . . . . . . . . . . 13 ((π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃) ↔ (βˆ€π‘Ž ∈ π‘Ÿ π‘Ž ∈ 𝑃 ∨ βˆ€π‘ ∈ 𝑠 𝑏 ∈ 𝑃))
2117, 20sylbb2 237 . . . . . . . . . . . 12 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))
2216, 21syl 17 . . . . . . . . . . 11 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 ∧ ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))
2314, 22sylbir 234 . . . . . . . . . 10 ((βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 ∧ βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))
2423expcom 413 . . . . . . . . 9 (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))
2513, 24syl6 35 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idlβ€˜π‘…)) ∧ (π‘Ÿ ∈ (Idlβ€˜π‘…) ∧ 𝑠 ∈ (Idlβ€˜π‘…))) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ (βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))))
2625ralrimdvva 3203 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (Idlβ€˜π‘…)) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))))
2726ex 412 . . . . . 6 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (Idlβ€˜π‘…) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))))
2827adantrd 491 . . . . 5 (𝑅 ∈ RingOps β†’ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) β†’ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))))
2928imdistand 570 . . . 4 (𝑅 ∈ RingOps β†’ (((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋) ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋) ∧ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))))
30 df-3an 1086 . . . 4 ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) ↔ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋) ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
31 df-3an 1086 . . . 4 ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))) ↔ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋) ∧ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃))))
3229, 30, 313imtr4g 296 . . 3 (𝑅 ∈ RingOps β†’ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))))
33 ispridl2.2 . . . 4 𝐻 = (2nd β€˜π‘…)
341, 33, 2ispridl 37415 . . 3 (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ÿ ∈ (Idlβ€˜π‘…)βˆ€π‘  ∈ (Idlβ€˜π‘…)(βˆ€π‘Ž ∈ π‘Ÿ βˆ€π‘ ∈ 𝑠 (π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ÿ βŠ† 𝑃 ∨ 𝑠 βŠ† 𝑃)))))
3532, 34sylibrd 259 . 2 (𝑅 ∈ RingOps β†’ ((𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ 𝑃 ∈ (PrIdlβ€˜π‘…)))
3635imp 406 1 ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) β†’ 𝑃 ∈ (PrIdlβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∨ wo 844   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055   βŠ† wss 3943  ran crn 5670  β€˜cfv 6537  (class class class)co 7405  1st c1st 7972  2nd c2nd 7973  RingOpscrngo 37275  Idlcidl 37388  PrIdlcpridl 37389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-idl 37391  df-pridl 37392
This theorem is referenced by:  ispridlc  37451
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