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Theorem ply1mulgsumlem1 47020
Description: Lemma 1 for ply1mulgsum 47024. (Contributed by AV, 19-Oct-2019.)
Hypotheses
Ref Expression
ply1mulgsum.p 𝑃 = (Poly1𝑅)
ply1mulgsum.b 𝐵 = (Base‘𝑃)
ply1mulgsum.a 𝐴 = (coe1𝐾)
ply1mulgsum.c 𝐶 = (coe1𝐿)
ply1mulgsum.x 𝑋 = (var1𝑅)
ply1mulgsum.pm × = (.r𝑃)
ply1mulgsum.sm · = ( ·𝑠𝑃)
ply1mulgsum.rm = (.r𝑅)
ply1mulgsum.m 𝑀 = (mulGrp‘𝑃)
ply1mulgsum.e = (.g𝑀)
Assertion
Ref Expression
ply1mulgsumlem1 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))
Distinct variable groups:   𝐴,𝑛,𝑠   𝐵,𝑛,𝑠   𝐶,𝑛,𝑠   𝑛,𝐾,𝑠   𝑛,𝐿,𝑠   𝑅,𝑛,𝑠
Allowed substitution hints:   𝑃(𝑛,𝑠)   · (𝑛,𝑠)   × (𝑛,𝑠)   (𝑛,𝑠)   (𝑛,𝑠)   𝑀(𝑛,𝑠)   𝑋(𝑛,𝑠)

Proof of Theorem ply1mulgsumlem1
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ply1mulgsum.a . . . 4 𝐴 = (coe1𝐾)
2 ply1mulgsum.b . . . 4 𝐵 = (Base‘𝑃)
3 ply1mulgsum.p . . . 4 𝑃 = (Poly1𝑅)
4 eqid 2732 . . . 4 (0g𝑅) = (0g𝑅)
51, 2, 3, 4coe1ae0 21731 . . 3 (𝐾𝐵 → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
653ad2ant2 1134 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
7 ply1mulgsum.c . . . . 5 𝐶 = (coe1𝐿)
87, 2, 3, 4coe1ae0 21731 . . . 4 (𝐿𝐵 → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
983ad2ant3 1135 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
10 nn0addcl 12503 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
1110adantr 481 . . . . . . . . . . . . . . 15 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
1211adantr 481 . . . . . . . . . . . . . 14 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → (𝑎 + 𝑏) ∈ ℕ0)
13 breq1 5150 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
1413imbi1d 341 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1514ralbidv 3177 . . . . . . . . . . . . . . 15 (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1615adantl 482 . . . . . . . . . . . . . 14 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) ∧ 𝑠 = (𝑎 + 𝑏)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
17 r19.26 3111 . . . . . . . . . . . . . . . 16 (∀𝑛 ∈ ℕ0 ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) ↔ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))))
18 nn0cn 12478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑎 ∈ ℕ0𝑎 ∈ ℂ)
1918adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20 nn0cn 12478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℕ0𝑏 ∈ ℂ)
2120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
2219, 21addcomd 11412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23223adant3 1132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
2423breq1d 5157 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25 nn0sumltlt 46979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛𝑎 < 𝑛))
2624, 25sylbid 239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
27263expia 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2827ancoms 459 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2928adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
3029imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
3130imim1d 82 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐶𝑛) = (0g𝑅))))
3231com23 86 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (𝐶𝑛) = (0g𝑅))))
3332imp 407 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (𝐶𝑛) = (0g𝑅)))
34 nn0sumltlt 46979 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
35343expia 1121 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3635adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3736imp 407 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
3837imim1d 82 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ((𝑎 + 𝑏) < 𝑛 → (𝐴𝑛) = (0g𝑅))))
3938com23 86 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → ((𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → (𝐴𝑛) = (0g𝑅))))
4039imp 407 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → (𝐴𝑛) = (0g𝑅)))
4133, 40anim12d 609 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → (((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ((𝐶𝑛) = (0g𝑅) ∧ (𝐴𝑛) = (0g𝑅))))
4241imp 407 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → ((𝐶𝑛) = (0g𝑅) ∧ (𝐴𝑛) = (0g𝑅)))
4342ancomd 462 . . . . . . . . . . . . . . . . . . 19 ((((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) ∧ ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))
4443exp31 420 . . . . . . . . . . . . . . . . . 18 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 → (((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
4544com23 86 . . . . . . . . . . . . . . . . 17 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
4645ralimdva 3167 . . . . . . . . . . . . . . . 16 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (∀𝑛 ∈ ℕ0 ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
4717, 46biimtrrid 242 . . . . . . . . . . . . . . 15 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
4847imp 407 . . . . . . . . . . . . . 14 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))
4912, 16, 48rspcedvd 3614 . . . . . . . . . . . . 13 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))
5049exp31 420 . . . . . . . . . . . 12 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
5150com23 86 . . . . . . . . . . 11 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → ((∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
5251expd 416 . . . . . . . . . 10 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))))
5352com34 91 . . . . . . . . 9 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))))
5453impancom 452 . . . . . . . 8 ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅))) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))))
5554com14 96 . . . . . . 7 (∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → (𝑏 ∈ ℕ0 → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅))) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))))
5655impcom 408 . . . . . 6 ((𝑏 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅))) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
5756rexlimiva 3147 . . . . 5 (∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅))) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
5857com13 88 . . . 4 ((𝑎 ∈ ℕ0 ∧ ∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅))) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
5958rexlimiva 3147 . . 3 (∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))))
609, 59mpcom 38 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → (∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
616, 60mpd 15 1 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wrex 3070   class class class wbr 5147  cfv 6540  (class class class)co 7405  cc 11104   + caddc 11109   < clt 11244  0cn0 12468  Basecbs 17140  .rcmulr 17194   ·𝑠 cvsca 17197  0gc0g 17381  .gcmg 18944  mulGrpcmgp 19981  Ringcrg 20049  var1cv1 21691  Poly1cpl1 21692  coe1cco1 21693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-tset 17212  df-ple 17213  df-psr 21453  df-mpl 21455  df-opsr 21457  df-psr1 21695  df-ply1 21697  df-coe1 21698
This theorem is referenced by:  ply1mulgsumlem2  47021
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