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Theorem ply1mulgsumlem1 44272
Description: Lemma 1 for ply1mulgsum 44276. (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 2826 . . . 4 (0g𝑅) = (0g𝑅)
51, 2, 3, 4coe1ae0 20301 . . 3 (𝐾𝐵 → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
653ad2ant2 1128 . 2 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑏 ∈ ℕ0𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))
7 ply1mulgsum.c . . . . 5 𝐶 = (coe1𝐿)
87, 2, 3, 4coe1ae0 20301 . . . 4 (𝐿𝐵 → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
983ad2ant3 1129 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
10 nn0addcl 11921 . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
1413imbi1d 343 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1514ralbidv 3202 . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . . . . 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 11896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑎 ∈ ℕ0𝑎 ∈ ℂ)
1918adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20 nn0cn 11896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℕ0𝑏 ∈ ℂ)
2120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
2219, 21addcomd 10831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23223adant3 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
2423breq1d 5073 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25 nn0sumltlt 44230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛𝑎 < 𝑛))
2624, 25sylbid 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
27263expia 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 44230 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
35343expia 1115 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . . . . . . . . . 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 3182 . . . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . 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 3286 . . . . 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 3286 . . 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144   class class class wbr 5063  cfv 6352  (class class class)co 7148  cc 10524   + caddc 10529   < clt 10664  0cn0 11886  Basecbs 16473  .rcmulr 16556   ·𝑠 cvsca 16559  0gc0g 16703  .gcmg 18154  mulGrpcmgp 19159  Ringcrg 19217  var1cv1 20261  Poly1cpl1 20262  coe1cco1 20263
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-tset 16574  df-ple 16575  df-psr 20055  df-mpl 20057  df-opsr 20059  df-psr1 20265  df-ply1 20267  df-coe1 20268
This theorem is referenced by:  ply1mulgsumlem2  44273
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