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Theorem pf1ind 19633
 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 5616 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
2 df1o2 7518 . . . . . . . . 9 1𝑜 = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
4 fvex 6160 . . . . . . . . . 10 (Base‘𝑅) ∈ V
53, 4eqeltri 2700 . . . . . . . . 9 𝐵 ∈ V
6 0ex 4755 . . . . . . . . 9 ∅ ∈ V
7 eqid 2626 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))
82, 5, 6, 7mapsncnv 7849 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
98coeq2i 5247 . . . . . . 7 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))
102, 5, 6, 7mapsnf1o2 7850 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
11 f1ococnv2 6122 . . . . . . . 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 2674 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ( I ↾ 𝐵))
1413coeq2d 5249 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
151, 14syl5eq 2672 . . . 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 19628 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
19 fcoi1 6037 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2016, 18, 193syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2115, 20eqtrd 2660 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = 𝐴)
22 pf1ind.cp . . . 4 + = (+g𝑅)
23 pf1ind.ct . . . 4 · = (.r𝑅)
24 eqid 2626 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
2524, 3evlval 19438 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
2625rneqi 5316 . . . 4 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘𝐵)
27 an4 864 . . . . . 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 2626 . . . . . . . . . . . 12 ran (1𝑜 eval 𝑅) = ran (1𝑜 eval 𝑅)
2917, 3, 28mpfpf1 19629 . . . . . . . . . . 11 (𝑎 ∈ ran (1𝑜 eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
3017, 3, 28mpfpf1 19629 . . . . . . . . . . 11 (𝑏 ∈ ran (1𝑜 eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
31 vex 3194 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
32 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3331, 32elab 3338 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
34 eleq1 2692 . . . . . . . . . . . . . . . 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 740 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3736anbi1d 740 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
38 ovex 6633 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
39 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
4038, 39elab 3338 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
41 oveq1 6612 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔))
4241eleq1d 2688 . . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 3194 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
46 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4745, 46elab 3338 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
48 eleq1 2692 . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})))
5150anbi1d 740 . . . . . . . . . . . . 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 6613 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
5352eleq1d 2688 . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 451 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5756an4s 868 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5857expimpd 628 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5944, 54, 58vtocl2ga 3265 . . . . . . . . . . 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 494 . . . . . . . . . 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 454 . . . . . . . . 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 446 . . . . . . . 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 19447 . . . . . . . . . . . 12 (𝑎 ∈ ran (1𝑜 eval 𝑅) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
6463ad2antrl 763 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
65 ffn 6004 . . . . . . . . . . 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 19447 . . . . . . . . . . . 12 (𝑏 ∈ ran (1𝑜 eval 𝑅) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
6867ad2antll 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
69 ffn 6004 . . . . . . . . . . 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 2626 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1𝑜 × {𝑤})) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
722, 5, 6, 71mapsnf1o3 7851 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜)
73 f1of 6096 . . . . . . . . . . 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 ovex 6633 . . . . . . . . . . 11 (𝐵𝑚 1𝑜) ∈ V
7675a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (𝐵𝑚 1𝑜) ∈ V)
775a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝐵 ∈ V)
78 inidm 3805 . . . . . . . . . 10 ((𝐵𝑚 1𝑜) ∩ (𝐵𝑚 1𝑜)) = (𝐵𝑚 1𝑜)
7966, 70, 74, 76, 76, 77, 78ofco 6871 . . . . . . . . 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})))))
8079eleq1d 2688 . . . . . . . 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}))
8162, 80sylibrd 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}))
8281expimpd 628 . . . . . 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}))
8327, 82syl5bi 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}))
8483imp 445 . . . 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})
85 ovex 6633 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
86 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
8785, 86elab 3338 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
88 oveq1 6612 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔))
8988eleq1d 2688 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
9087, 89syl5bbr 274 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
9137, 90imbi12d 334 . . . . . . . . . . . 12 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝜏𝜂) ∧ 𝜑) → 𝜎) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓})))
92 oveq2 6613 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
9392eleq1d 2688 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
9451, 93imbi12d 334 . . . . . . . . . . . 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})))
95 pf1ind.mu . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
9695expcom 451 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9796an4s 868 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9897expimpd 628 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9991, 94, 98vtocl2ga 3265 . . . . . . . . . . 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}))
10029, 30, 99syl2an 494 . . . . . . . . . 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}))
101100expcomd 454 . . . . . . . . 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})))
102101impcom 446 . . . . . . . 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}))
10366, 70, 74, 76, 76, 77, 78ofco 6871 . . . . . . . . 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})))))
104103eleq1d 2688 . . . . . . . 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}))
105102, 104sylibrd 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}))
106105expimpd 628 . . . . . 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}))
10727, 106syl5bi 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}))
108107imp 445 . . . 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})
109 coeq1 5244 . . . . 5 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
110109eleq1d 2688 . . . 4 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5244 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
112111eleq1d 2688 . . . 4 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5244 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
114113eleq1d 2688 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5244 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
116115eleq1d 2688 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5244 . . . . 5 (𝑦 = (𝑎𝑓 + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
118117eleq1d 2688 . . . 4 (𝑦 = (𝑎𝑓 + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
119 coeq1 5244 . . . . 5 (𝑦 = (𝑎𝑓 · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
120119eleq1d 2688 . . . 4 (𝑦 = (𝑎𝑓 · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
121 coeq1 5244 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
122121eleq1d 2688 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
12317pf1rcl 19627 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12416, 123syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
125124adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
126 1on 7513 . . . . . . . . . . . 12 1𝑜 ∈ On
127 eqid 2626 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
128127mplassa 19368 . . . . . . . . . . . 12 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 mPoly 𝑅) ∈ AssAlg)
129126, 124, 128sylancr 694 . . . . . . . . . . 11 (𝜑 → (1𝑜 mPoly 𝑅) ∈ AssAlg)
130 eqid 2626 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
131 eqid 2626 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
132130, 131ply1ascl 19542 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1𝑜 mPoly 𝑅))
133 eqid 2626 . . . . . . . . . . . 12 (Scalar‘(1𝑜 mPoly 𝑅)) = (Scalar‘(1𝑜 mPoly 𝑅))
134132, 133asclrhm 19256 . . . . . . . . . . 11 ((1𝑜 mPoly 𝑅) ∈ AssAlg → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
135129, 134syl 17 . . . . . . . . . 10 (𝜑 → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
136126a1i 11 . . . . . . . . . . . 12 (𝜑 → 1𝑜 ∈ On)
137127, 136, 124mplsca 19359 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1𝑜 mPoly 𝑅)))
138137oveq1d 6620 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1𝑜 mPoly 𝑅)) = ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
139135, 138eleqtrrd 2707 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)))
140 eqid 2626 . . . . . . . . . 10 (Base‘(1𝑜 mPoly 𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
1413, 140rhmf 18642 . . . . . . . . 9 ((algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)) → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
142139, 141syl 17 . . . . . . . 8 (𝜑 → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
143142ffvelrnda 6316 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅)))
144 eqid 2626 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
145144, 24, 3, 127, 140evl1val 19607 . . . . . . 7 ((𝑅 ∈ CRing ∧ ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅))) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
146125, 143, 145syl2anc 692 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
147144, 130, 3, 131evl1sca 19612 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
148124, 147sylan 488 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1493ressid 15851 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
150125, 149syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
151150oveq2d 6621 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly 𝑅))
152151fveq2d 6154 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly 𝑅)))
153152, 132syl6eqr 2678 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
154153fveq1d 6152 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
155154fveq2d 6154 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
156 eqid 2626 . . . . . . . . 9 (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly (𝑅s 𝐵))
157 eqid 2626 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
158 eqid 2626 . . . . . . . . 9 (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly (𝑅s 𝐵)))
159126a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1𝑜 ∈ On)
160 crngring 18474 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1613subrgid 18698 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
162124, 160, 1613syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
163162adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
164 simpr 477 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16525, 156, 157, 3, 158, 159, 125, 163, 164evlssca 19436 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
166155, 165eqtr3d 2662 . . . . . . 7 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
167166coeq1d 5248 . . . . . 6 ((𝜑𝑎𝐵) → (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
168146, 148, 1673eqtr3d 2668 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
169 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
170 snex 4874 . . . . . . . . . 10 {𝑓} ∈ V
1715, 170xpex 6916 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
172 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
173171, 172elab 3338 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
174169, 173sylibr 224 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
175174ralrimiva 2965 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
176 sneq 4163 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
177176xpeq2d 5104 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
178177eleq1d 2688 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
179178rspccva 3299 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
180175, 179sylan 488 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
181168, 180eqeltrrd 2705 . . . 4 ((𝜑𝑎𝐵) → (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
182 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
183 resiexg 7050 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1845, 183ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
185 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
186184, 185elab 3338 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
187182, 186sylibr 224 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18813, 187eqeltrd 2704 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
189 el1o 7525 . . . . . . . . . 10 (𝑎 ∈ 1𝑜𝑎 = ∅)
190 fveq2 6150 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
191189, 190sylbi 207 . . . . . . . . 9 (𝑎 ∈ 1𝑜 → (𝑏𝑎) = (𝑏‘∅))
192191mpteq2dv 4710 . . . . . . . 8 (𝑎 ∈ 1𝑜 → (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)))
193192coeq1d 5248 . . . . . . 7 (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
194193eleq1d 2688 . . . . . 6 (𝑎 ∈ 1𝑜 → (((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
195188, 194syl5ibrcom 237 . . . . 5 (𝜑 → (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
196195imp 445 . . . 4 ((𝜑𝑎 ∈ 1𝑜) → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19717, 3, 28pf1mpf 19630 . . . . 5 (𝐴𝑄 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
19816, 197syl 17 . . . 4 (𝜑 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
1993, 22, 23, 26, 84, 108, 110, 112, 114, 116, 118, 120, 122, 181, 196, 198mpfind 19450 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
20021, 199eqeltrrd 2705 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
201 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
202201elabg 3339 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
20316, 202syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
204200, 203mpbid 222 1 (𝜑𝜌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1992  {cab 2612  ∀wral 2912  Vcvv 3191  ∅c0 3896  {csn 4153   ↦ cmpt 4678   I cid 4989   × cxp 5077  ◡ccnv 5078  ran crn 5080   ↾ cres 5081   ∘ ccom 5083  Oncon0 5685   Fn wfn 5845  ⟶wf 5846  –1-1-onto→wf1o 5849  ‘cfv 5850  (class class class)co 6605   ∘𝑓 cof 6849  1𝑜c1o 7499   ↑𝑚 cmap 7803  Basecbs 15776   ↾s cress 15777  +gcplusg 15857  .rcmulr 15858  Scalarcsca 15860  Ringcrg 18463  CRingccrg 18464   RingHom crh 18628  SubRingcsubrg 18692  AssAlgcasa 19223  algSccascl 19225   mPoly cmpl 19267   evalSub ces 19418   eval cevl 19419  Poly1cpl1 19461  eval1ce1 19593 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-ofr 6852  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-sup 8293  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-mulr 15871  df-sca 15873  df-vsca 15874  df-ip 15875  df-tset 15876  df-ple 15877  df-ds 15880  df-hom 15882  df-cco 15883  df-0g 16018  df-gsum 16019  df-prds 16024  df-pws 16026  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-sbg 17343  df-mulg 17457  df-subg 17507  df-ghm 17574  df-cntz 17666  df-cmn 18111  df-abl 18112  df-mgp 18406  df-ur 18418  df-srg 18422  df-ring 18465  df-cring 18466  df-rnghom 18631  df-subrg 18694  df-lmod 18781  df-lss 18847  df-lsp 18886  df-assa 19226  df-asp 19227  df-ascl 19228  df-psr 19270  df-mvr 19271  df-mpl 19272  df-opsr 19274  df-evls 19420  df-evl 19421  df-psr1 19464  df-ply1 19466  df-evl1 19595 This theorem is referenced by:  pl1cn  29775
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