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Theorem pf1ind 19992
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 5840 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
2 df1o2 7777 . . . . . . . . 9 1𝑜 = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
43fvexi 6389 . . . . . . . . 9 𝐵 ∈ V
5 0ex 4950 . . . . . . . . 9 ∅ ∈ V
6 eqid 2765 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))
72, 4, 5, 6mapsncnv 8109 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
87coeq2i 5451 . . . . . . 7 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))
92, 4, 5, 6mapsnf1o2 8110 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵
10 f1ococnv2 6346 . . . . . . . 8 ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)):(𝐵𝑚 1𝑜)–1-1-onto𝐵 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
119, 10mp1i 13 . . . . . . 7 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) = ( I ↾ 𝐵))
128, 11syl5eqr 2813 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ( I ↾ 𝐵))
1312coeq2d 5453 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
141, 13syl5eq 2811 . . . 4 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝐴 ∘ ( I ↾ 𝐵)))
15 pf1ind.a . . . . 5 (𝜑𝐴𝑄)
16 pf1ind.cq . . . . . 6 𝑄 = ran (eval1𝑅)
1716, 3pf1f 19987 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
18 fcoi1 6260 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
1915, 17, 183syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2014, 19eqtrd 2799 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = 𝐴)
21 pf1ind.cp . . . 4 + = (+g𝑅)
22 pf1ind.ct . . . 4 · = (.r𝑅)
23 eqid 2765 . . . . . 6 (1𝑜 eval 𝑅) = (1𝑜 eval 𝑅)
2423, 3evlval 19797 . . . . 5 (1𝑜 eval 𝑅) = ((1𝑜 evalSub 𝑅)‘𝐵)
2524rneqi 5520 . . . 4 ran (1𝑜 eval 𝑅) = ran ((1𝑜 evalSub 𝑅)‘𝐵)
26 an4 646 . . . . . 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})))
27 eqid 2765 . . . . . . . . . . . 12 ran (1𝑜 eval 𝑅) = ran (1𝑜 eval 𝑅)
2816, 3, 27mpfpf1 19988 . . . . . . . . . . 11 (𝑎 ∈ ran (1𝑜 eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
2916, 3, 27mpfpf1 19988 . . . . . . . . . . 11 (𝑏 ∈ ran (1𝑜 eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ 𝑄)
30 vex 3353 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
31 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3230, 31elab 3504 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
33 eleq1 2832 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓 ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
3432, 33syl5bbr 276 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
3534anbi1d 623 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3635anbi1d 623 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
37 ovex 6874 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
38 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
3937, 38elab 3504 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
40 oveq1 6849 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔))
4140eleq1d 2829 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4239, 41syl5bbr 276 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4336, 42imbi12d 335 . . . . . . . . . . . 12 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝜏𝜂) ∧ 𝜑) → 𝜁) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓})))
44 vex 3353 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
45 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4644, 45elab 3504 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
47 eleq1 2832 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑔 ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
4846, 47syl5bbr 276 . . . . . . . . . . . . . . 15 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
4948anbi2d 622 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})))
5049anbi1d 623 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}) ∧ 𝜑)))
51 oveq2 6850 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
5251eleq1d 2829 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
5350, 52imbi12d 335 . . . . . . . . . . . 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})))
54 pf1ind.ad . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜁)
5554expcom 402 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5655an4s 650 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5756expimpd 445 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5843, 53, 57vtocl2ga 3426 . . . . . . . . . . 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}))
5928, 29, 58syl2an 589 . . . . . . . . . 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}))
6059expcomd 406 . . . . . . . . 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})))
6160impcom 396 . . . . . . . 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}))
6225, 3mpff 19806 . . . . . . . . . . . 12 (𝑎 ∈ ran (1𝑜 eval 𝑅) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
6362ad2antrl 719 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑎:(𝐵𝑚 1𝑜)⟶𝐵)
6463ffnd 6224 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑎 Fn (𝐵𝑚 1𝑜))
6525, 3mpff 19806 . . . . . . . . . . . 12 (𝑏 ∈ ran (1𝑜 eval 𝑅) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
6665ad2antll 720 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑏:(𝐵𝑚 1𝑜)⟶𝐵)
6766ffnd 6224 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝑏 Fn (𝐵𝑚 1𝑜))
68 eqid 2765 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1𝑜 × {𝑤})) = (𝑤𝐵 ↦ (1𝑜 × {𝑤}))
692, 4, 5, 68mapsnf1o3 8111 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜)
70 f1of 6320 . . . . . . . . . . 11 ((𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1𝑜) → (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵⟶(𝐵𝑚 1𝑜))
7169, 70mp1i 13 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (𝑤𝐵 ↦ (1𝑜 × {𝑤})):𝐵⟶(𝐵𝑚 1𝑜))
72 ovexd 6876 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → (𝐵𝑚 1𝑜) ∈ V)
734a1i 11 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1𝑜 eval 𝑅) ∧ 𝑏 ∈ ran (1𝑜 eval 𝑅))) → 𝐵 ∈ V)
74 inidm 3982 . . . . . . . . . 10 ((𝐵𝑚 1𝑜) ∩ (𝐵𝑚 1𝑜)) = (𝐵𝑚 1𝑜)
7564, 67, 71, 72, 72, 73, 74ofco 7115 . . . . . . . . 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})))))
7675eleq1d 2829 . . . . . . . 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}))
7761, 76sylibrd 250 . . . . . . 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}))
7877expimpd 445 . . . . . 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}))
7926, 78syl5bi 233 . . . . 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}))
8079imp 395 . . . 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})
81 ovex 6874 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
82 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
8381, 82elab 3504 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
84 oveq1 6849 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝑓𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔))
8584eleq1d 2829 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
8683, 85syl5bbr 276 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
8736, 86imbi12d 335 . . . . . . . . . . . 12 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((((𝜏𝜂) ∧ 𝜑) → 𝜎) ↔ ((((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓})))
88 oveq2 6850 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))))
8988eleq1d 2829 . . . . . . . . . . . . 13 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤})))) ∈ {𝑥𝜓}))
9050, 89imbi12d 335 . . . . . . . . . . . 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})))
91 pf1ind.mu . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂))) → 𝜎)
9291expcom 402 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9392an4s 650 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9493expimpd 445 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9587, 90, 94vtocl2ga 3426 . . . . . . . . . . 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}))
9628, 29, 95syl2an 589 . . . . . . . . . 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}))
9796expcomd 406 . . . . . . . . 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})))
9897impcom 396 . . . . . . . 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}))
9964, 67, 71, 72, 72, 73, 74ofco 7115 . . . . . . . . 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})))))
10099eleq1d 2829 . . . . . . . 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}))
10198, 100sylibrd 250 . . . . . . 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}))
102101expimpd 445 . . . . . 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}))
10326, 102syl5bi 233 . . . . 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}))
104103imp 395 . . . 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})
105 coeq1 5448 . . . . 5 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
106105eleq1d 2829 . . . 4 (𝑦 = ((𝐵𝑚 1𝑜) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
107 coeq1 5448 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
108107eleq1d 2829 . . . 4 (𝑦 = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
109 coeq1 5448 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
110109eleq1d 2829 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5448 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
112111eleq1d 2829 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5448 . . . . 5 (𝑦 = (𝑎𝑓 + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
114113eleq1d 2829 . . . 4 (𝑦 = (𝑎𝑓 + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5448 . . . . 5 (𝑦 = (𝑎𝑓 · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
116115eleq1d 2829 . . . 4 (𝑦 = (𝑎𝑓 · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5448 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
118117eleq1d 2829 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
11916pf1rcl 19986 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12015, 119syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
121120adantr 472 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
122 1on 7771 . . . . . . . . . . . 12 1𝑜 ∈ On
123 eqid 2765 . . . . . . . . . . . . 13 (1𝑜 mPoly 𝑅) = (1𝑜 mPoly 𝑅)
124123mplassa 19728 . . . . . . . . . . . 12 ((1𝑜 ∈ On ∧ 𝑅 ∈ CRing) → (1𝑜 mPoly 𝑅) ∈ AssAlg)
125122, 120, 124sylancr 581 . . . . . . . . . . 11 (𝜑 → (1𝑜 mPoly 𝑅) ∈ AssAlg)
126 eqid 2765 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
127 eqid 2765 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
128126, 127ply1ascl 19901 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1𝑜 mPoly 𝑅))
129 eqid 2765 . . . . . . . . . . . 12 (Scalar‘(1𝑜 mPoly 𝑅)) = (Scalar‘(1𝑜 mPoly 𝑅))
130128, 129asclrhm 19616 . . . . . . . . . . 11 ((1𝑜 mPoly 𝑅) ∈ AssAlg → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
131125, 130syl 17 . . . . . . . . . 10 (𝜑 → (algSc‘(Poly1𝑅)) ∈ ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
132122a1i 11 . . . . . . . . . . . 12 (𝜑 → 1𝑜 ∈ On)
133123, 132, 120mplsca 19719 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1𝑜 mPoly 𝑅)))
134133oveq1d 6857 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1𝑜 mPoly 𝑅)) = ((Scalar‘(1𝑜 mPoly 𝑅)) RingHom (1𝑜 mPoly 𝑅)))
135131, 134eleqtrrd 2847 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)))
136 eqid 2765 . . . . . . . . . 10 (Base‘(1𝑜 mPoly 𝑅)) = (Base‘(1𝑜 mPoly 𝑅))
1373, 136rhmf 18995 . . . . . . . . 9 ((algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1𝑜 mPoly 𝑅)) → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
138135, 137syl 17 . . . . . . . 8 (𝜑 → (algSc‘(Poly1𝑅)):𝐵⟶(Base‘(1𝑜 mPoly 𝑅)))
139138ffvelrnda 6549 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅)))
140 eqid 2765 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
141140, 23, 3, 123, 136evl1val 19966 . . . . . . 7 ((𝑅 ∈ CRing ∧ ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1𝑜 mPoly 𝑅))) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
142121, 139, 141syl2anc 579 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
143140, 126, 3, 127evl1sca 19971 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144120, 143sylan 575 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1453ressid 16209 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
146121, 145syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
147146oveq2d 6858 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly 𝑅))
148147fveq2d 6379 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly 𝑅)))
149148, 128syl6eqr 2817 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
150149fveq1d 6377 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
151150fveq2d 6379 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
152 eqid 2765 . . . . . . . . 9 (1𝑜 mPoly (𝑅s 𝐵)) = (1𝑜 mPoly (𝑅s 𝐵))
153 eqid 2765 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
154 eqid 2765 . . . . . . . . 9 (algSc‘(1𝑜 mPoly (𝑅s 𝐵))) = (algSc‘(1𝑜 mPoly (𝑅s 𝐵)))
155122a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1𝑜 ∈ On)
156 crngring 18825 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1573subrgid 19051 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158120, 156, 1573syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
159158adantr 472 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160 simpr 477 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16124, 152, 153, 3, 154, 155, 121, 159, 160evlssca 19795 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(1𝑜 mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
162151, 161eqtr3d 2801 . . . . . . 7 ((𝜑𝑎𝐵) → ((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵𝑚 1𝑜) × {𝑎}))
163162coeq1d 5452 . . . . . 6 ((𝜑𝑎𝐵) → (((1𝑜 eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
164142, 144, 1633eqtr3d 2807 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
165 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
166 snex 5064 . . . . . . . . . 10 {𝑓} ∈ V
1674, 166xpex 7160 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
168 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
169167, 168elab 3504 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
170165, 169sylibr 225 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
171170ralrimiva 3113 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
172 sneq 4344 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
173172xpeq2d 5307 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174173eleq1d 2829 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
175174rspccva 3460 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
176171, 175sylan 575 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
177164, 176eqeltrrd 2845 . . . 4 ((𝜑𝑎𝐵) → (((𝐵𝑚 1𝑜) × {𝑎}) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
178 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
179 resiexg 7300 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1804, 179ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
181 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
182180, 181elab 3504 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
183178, 182sylibr 225 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18412, 183eqeltrd 2844 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
185 el1o 7784 . . . . . . . . . 10 (𝑎 ∈ 1𝑜𝑎 = ∅)
186 fveq2 6375 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
187185, 186sylbi 208 . . . . . . . . 9 (𝑎 ∈ 1𝑜 → (𝑏𝑎) = (𝑏‘∅))
188187mpteq2dv 4904 . . . . . . . 8 (𝑎 ∈ 1𝑜 → (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)))
189188coeq1d 5452 . . . . . . 7 (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))))
190189eleq1d 2829 . . . . . 6 (𝑎 ∈ 1𝑜 → (((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
191184, 190syl5ibrcom 238 . . . . 5 (𝜑 → (𝑎 ∈ 1𝑜 → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓}))
192191imp 395 . . . 4 ((𝜑𝑎 ∈ 1𝑜) → ((𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19316, 3, 27pf1mpf 19989 . . . . 5 (𝐴𝑄 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
19415, 193syl 17 . . . 4 (𝜑 → (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∈ ran (1𝑜 eval 𝑅))
1953, 21, 22, 25, 80, 104, 106, 108, 110, 112, 114, 116, 118, 177, 192, 194mpfind 19809 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1𝑜) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1𝑜 × {𝑤}))) ∈ {𝑥𝜓})
19620, 195eqeltrrd 2845 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
197 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
198197elabg 3505 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
19915, 198syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
200196, 199mpbid 223 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  Vcvv 3350  c0 4079  {csn 4334  cmpt 4888   I cid 5184   × cxp 5275  ccnv 5276  ran crn 5278  cres 5279  ccom 5281  Oncon0 5908  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842  𝑓 cof 7093  1𝑜c1o 7757  𝑚 cmap 8060  Basecbs 16132  s cress 16133  +gcplusg 16216  .rcmulr 16217  Scalarcsca 16219  Ringcrg 18814  CRingccrg 18815   RingHom crh 18981  SubRingcsubrg 19045  AssAlgcasa 19583  algSccascl 19585   mPoly cmpl 19627   evalSub ces 19777   eval cevl 19778  Poly1cpl1 19820  eval1ce1 19952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-hom 16240  df-cco 16241  df-0g 16370  df-gsum 16371  df-prds 16376  df-pws 16378  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-mhm 17603  df-submnd 17604  df-grp 17694  df-minusg 17695  df-sbg 17696  df-mulg 17810  df-subg 17857  df-ghm 17924  df-cntz 18015  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-srg 18773  df-ring 18816  df-cring 18817  df-rnghom 18984  df-subrg 19047  df-lmod 19134  df-lss 19202  df-lsp 19244  df-assa 19586  df-asp 19587  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-evls 19779  df-evl 19780  df-psr1 19823  df-ply1 19825  df-evl1 19954
This theorem is referenced by:  pl1cn  30383
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