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Theorem ispridl2 36500
Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 36532 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 36478 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ π‘Ÿ ∈ (Idlβ€˜π‘…)) β†’ π‘Ÿ βŠ† 𝑋)
4 ssralv 4011 . . . . . . . . . . . . 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 716 . . . . . . . . . . 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 36478 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑠 ∈ (Idlβ€˜π‘…)) β†’ 𝑠 βŠ† 𝑋)
8 ssralv 4011 . . . . . . . . . . . . . 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 3167 . . . . . . . . . . . . 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 715 . . . . . . . . . . 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 714 . . . . . . . . 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 3136 . . . . . . . . . . 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 pm3.35 802 . . . . . . . . . . . . 13 (((π‘Žπ»π‘) ∈ 𝑃 ∧ ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))) β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))
16152ralimi 3127 . . . . . . . . . . . 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 3220 . . . . . . . . . . . . 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 3933 . . . . . . . . . . . . . 14 (π‘Ÿ βŠ† 𝑃 ↔ βˆ€π‘Ž ∈ π‘Ÿ π‘Ž ∈ 𝑃)
19 dfss3 3933 . . . . . . . . . . . . . 14 (𝑠 βŠ† 𝑃 ↔ βˆ€π‘ ∈ 𝑠 𝑏 ∈ 𝑃)
2018, 19orbi12i 914 . . . . . . . . . . . . 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 415 . . . . . . . . 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 3204 . . . . . . 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 414 . . . . . 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 493 . . . . 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 572 . . . 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 1090 . . . 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 1090 . . . 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 36496 . . 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 408 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 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065   βŠ† wss 3911  ran crn 5635  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  RingOpscrngo 36356  Idlcidl 36469  PrIdlcpridl 36470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-idl 36472  df-pridl 36473
This theorem is referenced by:  ispridlc  36532
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