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Theorem pf1ind 19659
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 5623 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
2 df1o2 7532 . . . . . . . . 9 1𝑜 = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
4 fvex 6168 . . . . . . . . . 10 (Base‘𝑅) ∈ V
53, 4eqeltri 2694 . . . . . . . . 9 𝐵 ∈ V
6 0ex 4760 . . . . . . . . 9 ∅ ∈ V
7 eqid 2621 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))
82, 5, 6, 7mapsncnv 7864 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
98coeq2i 5252 . . . . . . 7 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))
102, 5, 6, 7mapsnf1o2 7865 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
11 f1ococnv2 6130 . . . . . . . 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 2669 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ( I ↾ 𝐵))
1413coeq2d 5254 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
151, 14syl5eq 2667 . . . 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 19654 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
19 fcoi1 6045 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2016, 18, 193syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2115, 20eqtrd 2655 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = 𝐴)
22 pf1ind.cp . . . 4 + = (+g𝑅)
23 pf1ind.ct . . . 4 · = (.r𝑅)
24 eqid 2621 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
2524, 3evlval 19464 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
2625rneqi 5322 . . . 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 2621 . . . . . . . . . . . 12 ran (1𝑜 eval 𝑅) = ran (1𝑜 eval 𝑅)
2917, 3, 28mpfpf1 19655 . . . . . . . . . . 11 (𝑎 ∈ ran (1𝑜 eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
3017, 3, 28mpfpf1 19655 . . . . . . . . . . 11 (𝑏 ∈ ran (1𝑜 eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
31 vex 3193 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
32 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3331, 32elab 3338 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
34 eleq1 2686 . . . . . . . . . . . . . . . 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 6643 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
39 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
4038, 39elab 3338 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
41 oveq1 6622 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔))
4241eleq1d 2683 . . . . . . . . . . . . . 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 3193 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
46 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4745, 46elab 3338 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
48 eleq1 2686 . . . . . . . . . . . . . . . 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 6623 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
5352eleq1d 2683 . . . . . . . . . . . . 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 3264 . . . . . . . . . . 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 19473 . . . . . . . . . . . 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 6012 . . . . . . . . . . 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 19473 . . . . . . . . . . . 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 6012 . . . . . . . . . . 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 2621 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1𝑜 × {𝑤})) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
722, 5, 6, 71mapsnf1o3 7866 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜)
73 f1of 6104 . . . . . . . . . . 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 6645 . . . . . . . . . 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 3806 . . . . . . . . . 10 ((𝐵𝑚 1𝑜) ∩ (𝐵𝑚 1𝑜)) = (𝐵𝑚 1𝑜)
7866, 70, 74, 75, 75, 76, 77ofco 6882 . . . . . . . . 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 2683 . . . . . . . 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 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}))
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 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})
84 ovex 6643 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
85 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
8684, 85elab 3338 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
87 oveq1 6622 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔))
8887eleq1d 2683 . . . . . . . . . . . . . 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 334 . . . . . . . . . . . 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 6623 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
9291eleq1d 2683 . . . . . . . . . . . . 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 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})))
94 pf1ind.mu . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
9594expcom 451 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9695an4s 868 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9796expimpd 628 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9890, 93, 97vtocl2ga 3264 . . . . . . . . . . 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 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}))
10099expcomd 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})))
101100impcom 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}))
10266, 70, 74, 75, 75, 76, 77ofco 6882 . . . . . . . . 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 2683 . . . . . . . 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 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}))
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 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})
108 coeq1 5249 . . . . 5 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
109108eleq1d 2683 . . . 4 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
110 coeq1 5249 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
111110eleq1d 2683 . . . 4 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
112 coeq1 5249 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
113112eleq1d 2683 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
114 coeq1 5249 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
115114eleq1d 2683 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
116 coeq1 5249 . . . . 5 (𝑦 = (𝑎𝑓 + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
117116eleq1d 2683 . . . 4 (𝑦 = (𝑎𝑓 + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
118 coeq1 5249 . . . . 5 (𝑦 = (𝑎𝑓 · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
119118eleq1d 2683 . . . 4 (𝑦 = (𝑎𝑓 · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
120 coeq1 5249 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
121120eleq1d 2683 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
12217pf1rcl 19653 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12316, 122syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
124123adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
125 1on 7527 . . . . . . . . . . . 12 1𝑜 ∈ On
126 eqid 2621 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
127126mplassa 19394 . . . . . . . . . . . 12 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 mPoly 𝑅) ∈ AssAlg)
128125, 123, 127sylancr 694 . . . . . . . . . . 11 (𝜑 → (1𝑜 mPoly 𝑅) ∈ AssAlg)
129 eqid 2621 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
130 eqid 2621 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
131129, 130ply1ascl 19568 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1𝑜 mPoly 𝑅))
132 eqid 2621 . . . . . . . . . . . 12 (Scalar‘(1𝑜 mPoly 𝑅)) = (Scalar‘(1𝑜 mPoly 𝑅))
133131, 132asclrhm 19282 . . . . . . . . . . 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 19385 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1𝑜 mPoly 𝑅)))
137136oveq1d 6630 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1𝑜 mPoly 𝑅)) = ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
138134, 137eleqtrrd 2701 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)))
139 eqid 2621 . . . . . . . . . 10 (Base‘(1𝑜 mPoly 𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
1403, 139rhmf 18666 . . . . . . . . 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 6325 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅)))
143 eqid 2621 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
144143, 24, 3, 126, 139evl1val 19633 . . . . . . 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 692 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
146143, 129, 3, 130evl1sca 19638 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
147123, 146sylan 488 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1483ressid 15875 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
149124, 148syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
150149oveq2d 6631 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly 𝑅))
151150fveq2d 6162 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly 𝑅)))
152151, 131syl6eqr 2673 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
153152fveq1d 6160 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
154153fveq2d 6162 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
155 eqid 2621 . . . . . . . . 9 (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly (𝑅s 𝐵))
156 eqid 2621 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
157 eqid 2621 . . . . . . . . 9 (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly (𝑅s 𝐵)))
158125a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1𝑜 ∈ On)
159 crngring 18498 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1603subrgid 18722 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
161123, 159, 1603syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
162161adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
163 simpr 477 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16425, 155, 156, 3, 157, 158, 124, 162, 163evlssca 19462 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
165154, 164eqtr3d 2657 . . . . . . 7 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
166165coeq1d 5253 . . . . . 6 ((𝜑𝑎𝐵) → (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
167145, 147, 1663eqtr3d 2663 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
168 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
169 snex 4879 . . . . . . . . . 10 {𝑓} ∈ V
1705, 169xpex 6927 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
171 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
172170, 171elab 3338 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
173168, 172sylibr 224 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
174173ralrimiva 2962 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
175 sneq 4165 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
176175xpeq2d 5109 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
177176eleq1d 2683 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
178177rspccva 3298 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
179174, 178sylan 488 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
180167, 179eqeltrrd 2699 . . . 4 ((𝜑𝑎𝐵) → (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
181 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
182 resiexg 7064 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1835, 182ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
184 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
185183, 184elab 3338 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
186181, 185sylibr 224 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18713, 186eqeltrd 2698 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
188 el1o 7539 . . . . . . . . . 10 (𝑎 ∈ 1𝑜𝑎 = ∅)
189 fveq2 6158 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
190188, 189sylbi 207 . . . . . . . . 9 (𝑎 ∈ 1𝑜 → (𝑏𝑎) = (𝑏‘∅))
191190mpteq2dv 4715 . . . . . . . 8 (𝑎 ∈ 1𝑜 → (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)))
192191coeq1d 5253 . . . . . . 7 (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
193192eleq1d 2683 . . . . . 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 445 . . . 4 ((𝜑𝑎 ∈ 1𝑜) → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19617, 3, 28pf1mpf 19656 . . . . 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 19476 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19921, 198eqeltrrd 2699 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
200 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
201200elabg 3339 . . 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 384   = wceq 1480  wcel 1987  {cab 2607  wral 2908  Vcvv 3190  c0 3897  {csn 4155  cmpt 4683   I cid 4994   × cxp 5082  ccnv 5083  ran crn 5085  cres 5086  ccom 5088  Oncon0 5692   Fn wfn 5852  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  𝑓 cof 6860  1𝑜c1o 7513  𝑚 cmap 7817  Basecbs 15800  s cress 15801  +gcplusg 15881  .rcmulr 15882  Scalarcsca 15884  Ringcrg 18487  CRingccrg 18488   RingHom crh 18652  SubRingcsubrg 18716  AssAlgcasa 19249  algSccascl 19251   mPoly cmpl 19293   evalSub ces 19444   eval cevl 19445  Poly1cpl1 19487  eval1ce1 19619
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-sup 8308  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-hom 15906  df-cco 15907  df-0g 16042  df-gsum 16043  df-prds 16048  df-pws 16050  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-mulg 17481  df-subg 17531  df-ghm 17598  df-cntz 17690  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-srg 18446  df-ring 18489  df-cring 18490  df-rnghom 18655  df-subrg 18718  df-lmod 18805  df-lss 18873  df-lsp 18912  df-assa 19252  df-asp 19253  df-ascl 19254  df-psr 19296  df-mvr 19297  df-mpl 19298  df-opsr 19300  df-evls 19446  df-evl 19447  df-psr1 19490  df-ply1 19492  df-evl1 19621
This theorem is referenced by:  pl1cn  29825
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