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Theorem ply1mulgsumlem1 45308
Description: Lemma 1 for ply1mulgsum 45312. (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 2739 . . . 4 (0g𝑅) = (0g𝑅)
51, 2, 3, 4coe1ae0 21003 . . 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 21003 . . . 4 (𝐿𝐵 → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
983ad2ant3 1136 . . 3 ((𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵) → ∃𝑎 ∈ ℕ0𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)))
10 nn0addcl 12023 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 + 𝑏) ∈ ℕ0)
1110adantr 484 . . . . . . . . . . . . . . 15 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑎 + 𝑏) ∈ ℕ0)
1211adantr 484 . . . . . . . . . . . . . 14 ((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ (∀𝑛 ∈ ℕ0 (𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) ∧ ∀𝑛 ∈ ℕ0 (𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)))) → (𝑎 + 𝑏) ∈ ℕ0)
13 breq1 5043 . . . . . . . . . . . . . . . . 17 (𝑠 = (𝑎 + 𝑏) → (𝑠 < 𝑛 ↔ (𝑎 + 𝑏) < 𝑛))
1413imbi1d 345 . . . . . . . . . . . . . . . 16 (𝑠 = (𝑎 + 𝑏) → ((𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1514ralbidv 3110 . . . . . . . . . . . . . . 15 (𝑠 = (𝑎 + 𝑏) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅))) ↔ ∀𝑛 ∈ ℕ0 ((𝑎 + 𝑏) < 𝑛 → ((𝐴𝑛) = (0g𝑅) ∧ (𝐶𝑛) = (0g𝑅)))))
1615adantl 485 . . . . . . . . . . . . . 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 3085 . . . . . . . . . . . . . . . 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 11998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑎 ∈ ℕ0𝑎 ∈ ℂ)
1918adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑎 ∈ ℂ)
20 nn0cn 11998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑏 ∈ ℕ0𝑏 ∈ ℂ)
2120adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → 𝑏 ∈ ℂ)
2219, 21addcomd 10932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
23223adant3 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
2423breq1d 5050 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛 ↔ (𝑏 + 𝑎) < 𝑛))
25 nn0sumltlt 45267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑏 + 𝑎) < 𝑛𝑎 < 𝑛))
2624, 25sylbid 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛))
27263expia 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑏 ∈ ℕ0𝑎 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2827ancoms 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
2928adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑎 < 𝑛)))
3029imp 410 . . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑎 < 𝑛 → (𝐶𝑛) = (0g𝑅)) → (𝐶𝑛) = (0g𝑅)))
34 nn0sumltlt 45267 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑛 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛))
35343expia 1122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3635adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎 + 𝑏) < 𝑛𝑏 < 𝑛)))
3736imp 410 . . . . . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ (𝑅 ∈ Ring ∧ 𝐾𝐵𝐿𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ (𝑎 + 𝑏) < 𝑛) → ((𝑏 < 𝑛 → (𝐴𝑛) = (0g𝑅)) → (𝐴𝑛) = (0g𝑅)))
4133, 40anim12d 612 . . . . . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . 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 3092 . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . 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 3532 . . . . . . . . . . . . 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 423 . . . . . . . . . . . 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 419 . . . . . . . . . 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 411 . . . . . 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 3192 . . . . 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 3192 . . 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3054  wrex 3055   class class class wbr 5040  cfv 6349  (class class class)co 7182  cc 10625   + caddc 10630   < clt 10765  0cn0 11988  Basecbs 16598  .rcmulr 16681   ·𝑠 cvsca 16684  0gc0g 16828  .gcmg 18354  mulGrpcmgp 19370  Ringcrg 19428  var1cv1 20963  Poly1cpl1 20964  coe1cco1 20965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-iun 4893  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-of 7437  df-om 7612  df-1st 7726  df-2nd 7727  df-supp 7869  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-er 8332  df-map 8451  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-fsupp 8919  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-nn 11729  df-2 11791  df-3 11792  df-4 11793  df-5 11794  df-6 11795  df-7 11796  df-8 11797  df-9 11798  df-n0 11989  df-z 12075  df-dec 12192  df-uz 12337  df-fz 12994  df-struct 16600  df-ndx 16601  df-slot 16602  df-base 16604  df-sets 16605  df-ress 16606  df-plusg 16693  df-mulr 16694  df-sca 16696  df-vsca 16697  df-tset 16699  df-ple 16700  df-psr 20734  df-mpl 20736  df-opsr 20738  df-psr1 20967  df-ply1 20969  df-coe1 20970
This theorem is referenced by:  ply1mulgsumlem2  45309
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