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Theorem ply1mulgsumlem1 48303
Description: Lemma 1 for ply1mulgsum 48307. (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 2737 . . . 4 (0g𝑅) = (0g𝑅)
51, 2, 3, 4coe1ae0 22218 . . 3 (𝐾𝐵 → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
653ad2ant2 1135 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
7 ply1mulgsum.c . . . . 5 𝐶 = (coe1𝐿)
87, 2, 3, 4coe1ae0 22218 . . . 4 (𝐿𝐵 → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
983ad2ant3 1136 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
10 nn0addcl 12561 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
1110adantr 480 . . . . . . . . . . . . . . 15 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
1211adantr 480 . . . . . . . . . . . . . 14 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → (𝑎 + 𝑏) ∈ ℕ0)
13 breq1 5146 . . . . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . . . . 15 (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1615adantl 481 . . . . . . . . . . . . . 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 12536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑎 ∈ ℕ0𝑎 ∈ ℂ)
1918adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20 nn0cn 12536 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℕ0𝑏 ∈ ℂ)
2120adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
2219, 21addcomd 11463 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23223adant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
2423breq1d 5153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25 nn0sumltlt 48266 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛𝑎 < 𝑛))
2624, 25sylbid 240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
27263expia 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2827ancoms 458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2928adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
3029imp 406 . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (𝐶𝑛) = (0g𝑅)))
34 nn0sumltlt 48266 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
35343expia 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3635adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3736imp 406 . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . 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 243 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 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 3624 . . . . . . . . . . . . 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 419 . . . . . . . . . . . 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 415 . . . . . . . . . 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 451 . . . . . . . 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 407 . . . . . 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143  cfv 6561  (class class class)co 7431  cc 11153   + caddc 11158   < clt 11295  0cn0 12526  Basecbs 17247  .rcmulr 17298   ·𝑠 cvsca 17301  0gc0g 17484  .gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-ple 17317  df-psr 21929  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-ply1 22183  df-coe1 22184
This theorem is referenced by:  ply1mulgsumlem2  48304
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