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Theorem ply1mulgsumlem1 42702
Description: Lemma 1 for ply1mulgsum 42706. (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 2760 . . . 4 (0g𝑅) = (0g𝑅)
51, 2, 3, 4coe1ae0 19808 . . 3 (𝐾𝐵 → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
653ad2ant2 1129 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
7 ply1mulgsum.c . . . . 5 𝐶 = (coe1𝐿)
87, 2, 3, 4coe1ae0 19808 . . . 4 (𝐿𝐵 → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
983ad2ant3 1130 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
10 nn0addcl 11540 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
1110adantr 472 . . . . . . . . . . . . . . 15 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
1211adantr 472 . . . . . . . . . . . . . 14 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → (𝑎 + 𝑏) ∈ ℕ0)
13 breq1 4807 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
1413imbi1d 330 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1514ralbidv 3124 . . . . . . . . . . . . . . 15 (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1615adantl 473 . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . . . 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 11514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑎 ∈ ℕ0𝑎 ∈ ℂ)
1918adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20 nn0cn 11514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℕ0𝑏 ∈ ℂ)
2120adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
2219, 21addcomd 10450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23223adant3 1127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
2423breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25 nn0sumltlt 42656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛𝑎 < 𝑛))
2624, 25sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
27263expia 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2827ancoms 468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2928adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
3029imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (𝐶𝑛) = (0g𝑅)))
34 nn0sumltlt 42656 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
35343expia 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3635adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3736imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → (𝐴𝑛) = (0g𝑅)))
4133, 40anim12d 587 . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . 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 3100 . . . . . . . . . . . . . . . 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, 46syl5bir 233 . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . 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 3456 . . . . . . . . . . . . 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 631 . . . . . . . . . . . 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 451 . . . . . . . . . 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 455 . . . . . . . 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 445 . . . . . 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 3166 . . . . 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 3166 . . 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6814  cc 10146   + caddc 10151   < clt 10286  0cn0 11504  Basecbs 16079  .rcmulr 16164   ·𝑠 cvsca 16167  0gc0g 16322  .gcmg 17761  mulGrpcmgp 18709  Ringcrg 18767  var1cv1 19768  Poly1cpl1 19769  coe1cco1 19770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-fz 12540  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-sets 16086  df-ress 16087  df-plusg 16176  df-mulr 16177  df-sca 16179  df-vsca 16180  df-tset 16182  df-ple 16183  df-psr 19578  df-mpl 19580  df-opsr 19582  df-psr1 19772  df-ply1 19774  df-coe1 19775
This theorem is referenced by:  ply1mulgsumlem2  42703
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