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Theorem pf1ind 19913
Description: Prove a property of polynomials by "structural" induction, under a simplified model of structure which loses the sum of products structure. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pf1ind.cb 𝐵 = (Base‘𝑅)
pf1ind.cp + = (+g𝑅)
pf1ind.ct · = (.r𝑅)
pf1ind.cq 𝑄 = ran (eval1𝑅)
pf1ind.ad ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
pf1ind.mu ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
pf1ind.wa (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
pf1ind.wb (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
pf1ind.wc (𝑥 = 𝑓 → (𝜓𝜏))
pf1ind.wd (𝑥 = 𝑔 → (𝜓𝜂))
pf1ind.we (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
pf1ind.wf (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
pf1ind.wg (𝑥 = 𝐴 → (𝜓𝜌))
pf1ind.co ((𝜑𝑓𝐵) → 𝜒)
pf1ind.pr (𝜑𝜃)
pf1ind.a (𝜑𝐴𝑄)
Assertion
Ref Expression
pf1ind (𝜑𝜌)
Distinct variable groups:   𝑓,𝑔,𝑥, +   𝐵,𝑓,𝑔,𝑥   𝜂,𝑓,𝑥   𝜑,𝑓,𝑔   𝑥,𝐴   𝜒,𝑥   𝜓,𝑓,𝑔   𝑄,𝑓,𝑔   𝜌,𝑥   𝜎,𝑥   𝜏,𝑥   𝜃,𝑥   · ,𝑓,𝑔,𝑥   𝜁,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑓,𝑔)   𝜃(𝑓,𝑔)   𝜏(𝑓,𝑔)   𝜂(𝑔)   𝜁(𝑓,𝑔)   𝜎(𝑓,𝑔)   𝜌(𝑓,𝑔)   𝐴(𝑓,𝑔)   𝑄(𝑥)   𝑅(𝑥,𝑓,𝑔)

Proof of Theorem pf1ind
Dummy variables 𝑎 𝑏 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coass 5807 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
2 df1o2 7733 . . . . . . . . 9 1𝑜 = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
4 fvex 6354 . . . . . . . . . 10 (Base‘𝑅) ∈ V
53, 4eqeltri 2827 . . . . . . . . 9 𝐵 ∈ V
6 0ex 4934 . . . . . . . . 9 ∅ ∈ V
7 eqid 2752 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))
82, 5, 6, 7mapsncnv 8062 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
98coeq2i 5430 . . . . . . 7 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))
102, 5, 6, 7mapsnf1o2 8063 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
11 f1ococnv2 6316 . . . . . . . 8 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
1210, 11mp1i 13 . . . . . . 7 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
139, 12syl5eqr 2800 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ( I ↾ 𝐵))
1413coeq2d 5432 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
151, 14syl5eq 2798 . . . 4 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ( I ↾ 𝐵)))
16 pf1ind.a . . . . 5 (𝜑𝐴𝑄)
17 pf1ind.cq . . . . . 6 𝑄 = ran (eval1𝑅)
1817, 3pf1f 19908 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
19 fcoi1 6231 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2016, 18, 193syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2115, 20eqtrd 2786 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = 𝐴)
22 pf1ind.cp . . . 4 + = (+g𝑅)
23 pf1ind.ct . . . 4 · = (.r𝑅)
24 eqid 2752 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
2524, 3evlval 19718 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
2625rneqi 5499 . . . 4 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘𝐵)
27 an4 900 . . . . . 6 (((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ (𝑏 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})) ↔ ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})))
28 eqid 2752 . . . . . . . . . . . 12 ran (1𝑜 eval 𝑅) = ran (1𝑜 eval 𝑅)
2917, 3, 28mpfpf1 19909 . . . . . . . . . . 11 (𝑎 ∈ ran (1𝑜 eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
3017, 3, 28mpfpf1 19909 . . . . . . . . . . 11 (𝑏 ∈ ran (1𝑜 eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
31 vex 3335 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
32 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3331, 32elab 3482 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
34 eleq1 2819 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓 ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
3533, 34syl5bbr 274 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
3635anbi1d 743 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3736anbi1d 743 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
38 ovex 6833 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
39 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
4038, 39elab 3482 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
41 oveq1 6812 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔))
4241eleq1d 2816 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4340, 42syl5bbr 274 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4437, 43imbi12d 333 . . . . . . . . . . . 12 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝜏𝜂) ∧ 𝜑) → 𝜁) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓})))
45 vex 3335 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
46 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4745, 46elab 3482 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
48 eleq1 2819 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑔 ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
4947, 48syl5bbr 274 . . . . . . . . . . . . . . 15 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
5049anbi2d 742 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})))
5150anbi1d 743 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑)))
52 oveq2 6813 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
5352eleq1d 2816 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
5451, 53imbi12d 333 . . . . . . . . . . . 12 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓})))
55 pf1ind.ad . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
5655expcom 450 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5756an4s 904 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5857expimpd 630 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5944, 54, 58vtocl2ga 3406 . . . . . . . . . . 11 (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
6029, 30, 59syl2an 495 . . . . . . . . . 10 ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
6160expcomd 453 . . . . . . . . 9 ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓})))
6261impcom 445 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
6326, 3mpff 19727 . . . . . . . . . . . 12 (𝑎 ∈ ran (1𝑜 eval 𝑅) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
6463ad2antrl 766 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
65 ffn 6198 . . . . . . . . . . 11 (𝑎:(𝐵𝑚 1𝑜)⟶𝐵𝑎 Fn (𝐵𝑚 1𝑜))
6664, 65syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑎 Fn (𝐵𝑚 1𝑜))
6726, 3mpff 19727 . . . . . . . . . . . 12 (𝑏 ∈ ran (1𝑜 eval 𝑅) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
6867ad2antll 767 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
69 ffn 6198 . . . . . . . . . . 11 (𝑏:(𝐵𝑚 1𝑜)⟶𝐵𝑏 Fn (𝐵𝑚 1𝑜))
7068, 69syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑏 Fn (𝐵𝑚 1𝑜))
71 eqid 2752 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1𝑜 × {𝑤})) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
722, 5, 6, 71mapsnf1o3 8064 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜)
73 f1of 6290 . . . . . . . . . . 11 ((𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜) → (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵⟶(𝐵𝑚 1𝑜))
7472, 73mp1i 13 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵⟶(𝐵𝑚 1𝑜))
75 ovexd 6835 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (𝐵𝑚 1𝑜) ∈ V)
765a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝐵 ∈ V)
77 inidm 3957 . . . . . . . . . 10 ((𝐵𝑚 1𝑜) ∩ (𝐵𝑚 1𝑜)) = (𝐵𝑚 1𝑜)
7866, 70, 74, 75, 75, 76, 77ofco 7074 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
7978eleq1d 2816 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
8062, 79sylibrd 249 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
8180expimpd 630 . . . . . 6 (𝜑 → (((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})) → ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
8227, 81syl5bi 232 . . . . 5 (𝜑 → (((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ (𝑏 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})) → ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
8382imp 444 . . . 4 ((𝜑 ∧ ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ (𝑏 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))) → ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
84 ovex 6833 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
85 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
8684, 85elab 3482 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
87 oveq1 6812 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔))
8887eleq1d 2816 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
8986, 88syl5bbr 274 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
9037, 89imbi12d 333 . . . . . . . . . . . 12 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝜏𝜂) ∧ 𝜑) → 𝜎) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓})))
91 oveq2 6813 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
9291eleq1d 2816 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
9351, 92imbi12d 333 . . . . . . . . . . . 12 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓})))
94 pf1ind.mu . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
9594expcom 450 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9695an4s 904 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9796expimpd 630 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9890, 93, 97vtocl2ga 3406 . . . . . . . . . . 11 (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄 ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
9929, 30, 98syl2an 495 . . . . . . . . . 10 ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
10099expcomd 453 . . . . . . . . 9 ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) → (𝜑 → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓})))
101100impcom 445 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
10266, 70, 74, 75, 75, 76, 77ofco 7074 . . . . . . . . 9 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
103102eleq1d 2816 . . . . . . . 8 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
104101, 103sylibrd 249 . . . . . . 7 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) → ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
105104expimpd 630 . . . . . 6 (𝜑 → (((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅)) ∧ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})) → ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
10627, 105syl5bi 232 . . . . 5 (𝜑 → (((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ (𝑏 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})) → ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
107106imp 444 . . . 4 ((𝜑 ∧ ((𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ (𝑏 ∈ ran (1𝑜 eval 𝑅) ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))) → ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
108 coeq1 5427 . . . . 5 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
109108eleq1d 2816 . . . 4 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
110 coeq1 5427 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
111110eleq1d 2816 . . . 4 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
112 coeq1 5427 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
113112eleq1d 2816 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
114 coeq1 5427 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
115114eleq1d 2816 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
116 coeq1 5427 . . . . 5 (𝑦 = (𝑎𝑓 + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
117116eleq1d 2816 . . . 4 (𝑦 = (𝑎𝑓 + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
118 coeq1 5427 . . . . 5 (𝑦 = (𝑎𝑓 · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
119118eleq1d 2816 . . . 4 (𝑦 = (𝑎𝑓 · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
120 coeq1 5427 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
121120eleq1d 2816 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
12217pf1rcl 19907 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12316, 122syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
124123adantr 472 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
125 1on 7728 . . . . . . . . . . . 12 1𝑜 ∈ On
126 eqid 2752 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
127126mplassa 19648 . . . . . . . . . . . 12 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 mPoly 𝑅) ∈ AssAlg)
128125, 123, 127sylancr 698 . . . . . . . . . . 11 (𝜑 → (1𝑜 mPoly 𝑅) ∈ AssAlg)
129 eqid 2752 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
130 eqid 2752 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
131129, 130ply1ascl 19822 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1𝑜 mPoly 𝑅))
132 eqid 2752 . . . . . . . . . . . 12 (Scalar‘(1𝑜 mPoly 𝑅)) = (Scalar‘(1𝑜 mPoly 𝑅))
133131, 132asclrhm 19536 . . . . . . . . . . 11 ((1𝑜 mPoly 𝑅) ∈ AssAlg → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
134128, 133syl 17 . . . . . . . . . 10 (𝜑 → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
135125a1i 11 . . . . . . . . . . . 12 (𝜑 → 1𝑜 ∈ On)
136126, 135, 123mplsca 19639 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1𝑜 mPoly 𝑅)))
137136oveq1d 6820 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1𝑜 mPoly 𝑅)) = ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
138134, 137eleqtrrd 2834 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)))
139 eqid 2752 . . . . . . . . . 10 (Base‘(1𝑜 mPoly 𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
1403, 139rhmf 18920 . . . . . . . . 9 ((algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)) → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
141138, 140syl 17 . . . . . . . 8 (𝜑 → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
142141ffvelrnda 6514 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅)))
143 eqid 2752 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
144143, 24, 3, 126, 139evl1val 19887 . . . . . . 7 ((𝑅 ∈ CRing ∧ ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅))) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
145124, 142, 144syl2anc 696 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
146143, 129, 3, 130evl1sca 19892 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
147123, 146sylan 489 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1483ressid 16129 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
149124, 148syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
150149oveq2d 6821 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly 𝑅))
151150fveq2d 6348 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly 𝑅)))
152151, 131syl6eqr 2804 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
153152fveq1d 6346 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
154153fveq2d 6348 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
155 eqid 2752 . . . . . . . . 9 (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly (𝑅s 𝐵))
156 eqid 2752 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
157 eqid 2752 . . . . . . . . 9 (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly (𝑅s 𝐵)))
158125a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1𝑜 ∈ On)
159 crngring 18750 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1603subrgid 18976 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
161123, 159, 1603syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
162161adantr 472 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
163 simpr 479 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16425, 155, 156, 3, 157, 158, 124, 162, 163evlssca 19716 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
165154, 164eqtr3d 2788 . . . . . . 7 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
166165coeq1d 5431 . . . . . 6 ((𝜑𝑎𝐵) → (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
167145, 147, 1663eqtr3d 2794 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
168 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
169 snex 5049 . . . . . . . . . 10 {𝑓} ∈ V
1705, 169xpex 7119 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
171 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
172170, 171elab 3482 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
173168, 172sylibr 224 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
174173ralrimiva 3096 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
175 sneq 4323 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
176175xpeq2d 5288 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
177176eleq1d 2816 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
178177rspccva 3440 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
179174, 178sylan 489 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
180167, 179eqeltrrd 2832 . . . 4 ((𝜑𝑎𝐵) → (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
181 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
182 resiexg 7259 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1835, 182ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
184 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
185183, 184elab 3482 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
186181, 185sylibr 224 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18713, 186eqeltrd 2831 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
188 el1o 7740 . . . . . . . . . 10 (𝑎 ∈ 1𝑜𝑎 = ∅)
189 fveq2 6344 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
190188, 189sylbi 207 . . . . . . . . 9 (𝑎 ∈ 1𝑜 → (𝑏𝑎) = (𝑏‘∅))
191190mpteq2dv 4889 . . . . . . . 8 (𝑎 ∈ 1𝑜 → (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)))
192191coeq1d 5431 . . . . . . 7 (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
193192eleq1d 2816 . . . . . 6 (𝑎 ∈ 1𝑜 → (((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
194187, 193syl5ibrcom 237 . . . . 5 (𝜑 → (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
195194imp 444 . . . 4 ((𝜑𝑎 ∈ 1𝑜) → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19617, 3, 28pf1mpf 19910 . . . . 5 (𝐴𝑄 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
19716, 196syl 17 . . . 4 (𝜑 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
1983, 22, 23, 26, 83, 107, 109, 111, 113, 115, 117, 119, 121, 180, 195, 197mpfind 19730 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19921, 198eqeltrrd 2832 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
200 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
201200elabg 3483 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
20216, 201syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
203199, 202mpbid 222 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  {cab 2738  wral 3042  Vcvv 3332  c0 4050  {csn 4313  cmpt 4873   I cid 5165   × cxp 5256  ccnv 5257  ran crn 5259  cres 5260  ccom 5262  Oncon0 5876   Fn wfn 6036  wf 6037  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805  𝑓 cof 7052  1𝑜c1o 7714  𝑚 cmap 8015  Basecbs 16051  s cress 16052  +gcplusg 16135  .rcmulr 16136  Scalarcsca 16138  Ringcrg 18739  CRingccrg 18740   RingHom crh 18906  SubRingcsubrg 18970  AssAlgcasa 19503  algSccascl 19505   mPoly cmpl 19547   evalSub ces 19698   eval cevl 19699  Poly1cpl1 19741  eval1ce1 19873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-ofr 7055  df-om 7223  df-1st 7325  df-2nd 7326  df-supp 7456  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8433  df-sup 8505  df-oi 8572  df-card 8947  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-fzo 12652  df-seq 12988  df-hash 13304  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-mulr 16149  df-sca 16151  df-vsca 16152  df-ip 16153  df-tset 16154  df-ple 16155  df-ds 16158  df-hom 16160  df-cco 16161  df-0g 16296  df-gsum 16297  df-prds 16302  df-pws 16304  df-mre 16440  df-mrc 16441  df-acs 16443  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-mhm 17528  df-submnd 17529  df-grp 17618  df-minusg 17619  df-sbg 17620  df-mulg 17734  df-subg 17784  df-ghm 17851  df-cntz 17942  df-cmn 18387  df-abl 18388  df-mgp 18682  df-ur 18694  df-srg 18698  df-ring 18741  df-cring 18742  df-rnghom 18909  df-subrg 18972  df-lmod 19059  df-lss 19127  df-lsp 19166  df-assa 19506  df-asp 19507  df-ascl 19508  df-psr 19550  df-mvr 19551  df-mpl 19552  df-opsr 19554  df-evls 19700  df-evl 19701  df-psr1 19744  df-ply1 19746  df-evl1 19875
This theorem is referenced by:  pl1cn  30302
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