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Theorem pf1ind 20204
 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 6000 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
2 df1o2 7974 . . . . . . . . 9 1o = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
43fvexi 6559 . . . . . . . . 9 𝐵 ∈ V
5 0ex 5109 . . . . . . . . 9 ∅ ∈ V
6 eqid 2797 . . . . . . . . 9 (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))
72, 4, 5, 6mapsncnv 8313 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1o × {𝑤}))
87coeq2i 5624 . . . . . . 7 ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))
92, 4, 5, 6mapsnf1o2 8314 . . . . . . . 8 (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)):(𝐵𝑚 1o)–1-1-onto𝐵
10 f1ococnv2 6516 . . . . . . . 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 2847 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
1312coeq2d 5626 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
141, 13syl5eq 2845 . . . 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 20199 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
18 fcoi1 6427 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
1915, 17, 183syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2014, 19eqtrd 2833 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = 𝐴)
21 pf1ind.cp . . . 4 + = (+g𝑅)
22 pf1ind.ct . . . 4 · = (.r𝑅)
23 eqid 2797 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
2423, 3evlval 19995 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
2524rneqi 5696 . . . 4 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26 an4 652 . . . . . 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 2797 . . . . . . . . . . . 12 ran (1o eval 𝑅) = ran (1o eval 𝑅)
2816, 3, 27mpfpf1 20200 . . . . . . . . . . 11 (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
2916, 3, 27mpfpf1 20200 . . . . . . . . . . 11 (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30 vex 3443 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
31 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3230, 31elab 3608 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
33 eleq1 2872 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3432, 33syl5bbr 286 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3534anbi1d 629 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3635anbi1d 629 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
37 ovex 7055 . . . . . . . . . . . . . . 15 (𝑓𝑓 + 𝑔) ∈ V
38 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 + 𝑔) → (𝜓𝜁))
3937, 38elab 3608 . . . . . . . . . . . . . 14 ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
40 oveq1 7030 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 + 𝑔))
4140eleq1d 2869 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓𝑓 + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4239, 41syl5bbr 286 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 + 𝑔) ∈ {𝑥𝜓}))
4336, 42imbi12d 346 . . . . . . . . . . . 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 3443 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
45 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4644, 45elab 3608 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
47 eleq1 2872 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4846, 47syl5bbr 286 . . . . . . . . . . . . . . 15 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4948anbi2d 628 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})))
5049anbi1d 629 . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 + (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
5251eleq1d 2869 . . . . . . . . . . . . 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 346 . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5655an4s 656 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5756expimpd 454 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5843, 53, 57vtocl2ga 3520 . . . . . . . . . . 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 595 . . . . . . . . . 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 417 . . . . . . . . 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 408 . . . . . . . 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 20004 . . . . . . . . . . . 12 (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵𝑚 1o)⟶𝐵)
6362ad2antrl 724 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵𝑚 1o)⟶𝐵)
6463ffnd 6390 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵𝑚 1o))
6525, 3mpff 20004 . . . . . . . . . . . 12 (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵𝑚 1o)⟶𝐵)
6665ad2antll 725 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵𝑚 1o)⟶𝐵)
6766ffnd 6390 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵𝑚 1o))
68 eqid 2797 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1o × {𝑤})) = (𝑤𝐵 ↦ (1o × {𝑤}))
692, 4, 5, 68mapsnf1o3 8315 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1o × {𝑤})):𝐵1-1-onto→(𝐵𝑚 1o)
70 f1of 6490 . . . . . . . . . . 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 7057 . . . . . . . . . 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 4121 . . . . . . . . . 10 ((𝐵𝑚 1o) ∩ (𝐵𝑚 1o)) = (𝐵𝑚 1o)
7564, 67, 71, 72, 72, 73, 74ofco 7294 . . . . . . . . 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 2869 . . . . . . . 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 260 . . . . . . 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 454 . . . . . 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 243 . . . . 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 407 . . . 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 7055 . . . . . . . . . . . . . . 15 (𝑓𝑓 · 𝑔) ∈ V
82 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓𝑓 · 𝑔) → (𝜓𝜎))
8381, 82elab 3608 . . . . . . . . . . . . . 14 ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
84 oveq1 7030 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 · 𝑔))
8584eleq1d 2869 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓𝑓 · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
8683, 85syl5bbr 286 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 · 𝑔) ∈ {𝑥𝜓}))
8736, 86imbi12d 346 . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘𝑓 · (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
8988eleq1d 2869 . . . . . . . . . . . . 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 346 . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9392an4s 656 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9493expimpd 454 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9587, 90, 94vtocl2ga 3520 . . . . . . . . . . 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 595 . . . . . . . . . 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 417 . . . . . . . . 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 408 . . . . . . . 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 7294 . . . . . . . . 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 2869 . . . . . . . 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 260 . . . . . . 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 454 . . . . . 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 243 . . . . 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 407 . . . 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 5621 . . . . 5 (𝑦 = ((𝐵𝑚 1o) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵𝑚 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
106105eleq1d 2869 . . . 4 (𝑦 = ((𝐵𝑚 1o) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵𝑚 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
107 coeq1 5621 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
108107eleq1d 2869 . . . 4 (𝑦 = (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
109 coeq1 5621 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
110109eleq1d 2869 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5621 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
112111eleq1d 2869 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5621 . . . . 5 (𝑦 = (𝑎𝑓 + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
114113eleq1d 2869 . . . 4 (𝑦 = (𝑎𝑓 + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5621 . . . . 5 (𝑦 = (𝑎𝑓 · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
116115eleq1d 2869 . . . 4 (𝑦 = (𝑎𝑓 · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎𝑓 · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5621 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
118117eleq1d 2869 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
11916pf1rcl 20198 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12015, 119syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
121120adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
122 1on 7967 . . . . . . . . . . . 12 1o ∈ On
123 eqid 2797 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
124123mplassa 19926 . . . . . . . . . . . 12 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125122, 120, 124sylancr 587 . . . . . . . . . . 11 (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126 eqid 2797 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
127 eqid 2797 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
128126, 127ply1ascl 20113 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1o mPoly 𝑅))
129 eqid 2797 . . . . . . . . . . . 12 (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130128, 129asclrhm 19811 . . . . . . . . . . 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 19917 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1o mPoly 𝑅)))
134133oveq1d 7038 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135131, 134eleqtrrd 2888 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136 eqid 2797 . . . . . . . . . 10 (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
1373, 136rhmf 19172 . . . . . . . . 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 6723 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140 eqid 2797 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
141140, 23, 3, 123, 136evl1val 20178 . . . . . . 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 584 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
143140, 126, 3, 127evl1sca 20183 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144120, 143sylan 580 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1453ressid 16392 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
146121, 145syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
147146oveq2d 7039 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1o mPoly (𝑅s 𝐵)) = (1o mPoly 𝑅))
148147fveq2d 6549 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149148, 128syl6eqr 2851 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
150149fveq1d 6547 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
151150fveq2d 6549 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
152 eqid 2797 . . . . . . . . 9 (1o mPoly (𝑅s 𝐵)) = (1o mPoly (𝑅s 𝐵))
153 eqid 2797 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
154 eqid 2797 . . . . . . . . 9 (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly (𝑅s 𝐵)))
155122a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1o ∈ On)
156 crngring 19002 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1573subrgid 19231 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158120, 156, 1573syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
159158adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160 simpr 485 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16124, 152, 153, 3, 154, 155, 121, 159, 160evlssca 19993 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵𝑚 1o) × {𝑎}))
162151, 161eqtr3d 2835 . . . . . . 7 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵𝑚 1o) × {𝑎}))
163162coeq1d 5625 . . . . . 6 ((𝜑𝑎𝐵) → (((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵𝑚 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
164142, 144, 1633eqtr3d 2841 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵𝑚 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
165 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
166 snex 5230 . . . . . . . . . 10 {𝑓} ∈ V
1674, 166xpex 7340 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
168 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
169167, 168elab 3608 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
170165, 169sylibr 235 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
171170ralrimiva 3151 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
172 sneq 4488 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
173172xpeq2d 5480 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174173eleq1d 2869 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
175174rspccva 3560 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
176171, 175sylan 580 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
177164, 176eqeltrrd 2886 . . . 4 ((𝜑𝑎𝐵) → (((𝐵𝑚 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
178 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
179 resiexg 7482 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1804, 179ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
181 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
182180, 181elab 3608 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
183178, 182sylibr 235 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18412, 183eqeltrd 2885 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
185 el1o 7982 . . . . . . . . . 10 (𝑎 ∈ 1o𝑎 = ∅)
186 fveq2 6545 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
187185, 186sylbi 218 . . . . . . . . 9 (𝑎 ∈ 1o → (𝑏𝑎) = (𝑏‘∅))
188187mpteq2dv 5063 . . . . . . . 8 (𝑎 ∈ 1o → (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)))
189188coeq1d 5625 . . . . . . 7 (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
190189eleq1d 2869 . . . . . 6 (𝑎 ∈ 1o → (((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
191184, 190syl5ibrcom 248 . . . . 5 (𝜑 → (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
192191imp 407 . . . 4 ((𝜑𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
19316, 3, 27pf1mpf 20201 . . . . 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 20007 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵𝑚 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
19620, 195eqeltrrd 2886 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
197 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
198197elabg 3607 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
19915, 198syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
200196, 199mpbid 233 1 (𝜑𝜌)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  {cab 2777  ∀wral 3107  Vcvv 3440  ∅c0 4217  {csn 4478   ↦ cmpt 5047   I cid 5354   × cxp 5448  ◡ccnv 5449  ran crn 5451   ↾ cres 5452   ∘ ccom 5454  Oncon0 6073  ⟶wf 6228  –1-1-onto→wf1o 6231  ‘cfv 6232  (class class class)co 7023   ∘𝑓 cof 7272  1oc1o 7953   ↑𝑚 cmap 8263  Basecbs 16316   ↾s cress 16317  +gcplusg 16398  .rcmulr 16399  Scalarcsca 16401  Ringcrg 18991  CRingccrg 18992   RingHom crh 19158  SubRingcsubrg 19225  AssAlgcasa 19775  algSccascl 19777   mPoly cmpl 19825   evalSub ces 19975   eval cevl 19976  Poly1cpl1 20032  eval1ce1 20164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-ofr 7275  df-om 7444  df-1st 7552  df-2nd 7553  df-supp 7689  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-2o 7961  df-oadd 7964  df-er 8146  df-map 8265  df-pm 8266  df-ixp 8318  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-fsupp 8687  df-sup 8759  df-oi 8827  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-2 11554  df-3 11555  df-4 11556  df-5 11557  df-6 11558  df-7 11559  df-8 11560  df-9 11561  df-n0 11752  df-z 11836  df-dec 11953  df-uz 12098  df-fz 12747  df-fzo 12888  df-seq 13224  df-hash 13545  df-struct 16318  df-ndx 16319  df-slot 16320  df-base 16322  df-sets 16323  df-ress 16324  df-plusg 16411  df-mulr 16412  df-sca 16414  df-vsca 16415  df-ip 16416  df-tset 16417  df-ple 16418  df-ds 16420  df-hom 16422  df-cco 16423  df-0g 16548  df-gsum 16549  df-prds 16554  df-pws 16556  df-mre 16690  df-mrc 16691  df-acs 16693  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-mhm 17778  df-submnd 17779  df-grp 17868  df-minusg 17869  df-sbg 17870  df-mulg 17986  df-subg 18034  df-ghm 18101  df-cntz 18192  df-cmn 18639  df-abl 18640  df-mgp 18934  df-ur 18946  df-srg 18950  df-ring 18993  df-cring 18994  df-rnghom 19161  df-subrg 19227  df-lmod 19330  df-lss 19398  df-lsp 19438  df-assa 19778  df-asp 19779  df-ascl 19780  df-psr 19828  df-mvr 19829  df-mpl 19830  df-opsr 19832  df-evls 19977  df-evl 19978  df-psr1 20035  df-ply1 20037  df-evl1 20166 This theorem is referenced by:  pl1cn  30811
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