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Theorem pf1ind 22359
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 6285 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
2 df1o2 8513 . . . . . . . . 9 1o = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
43fvexi 6920 . . . . . . . . 9 𝐵 ∈ V
5 0ex 5307 . . . . . . . . 9 ∅ ∈ V
6 eqid 2737 . . . . . . . . 9 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))
72, 4, 5, 6mapsncnv 8933 . . . . . . . 8 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1o × {𝑤}))
87coeq2i 5871 . . . . . . 7 ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))
92, 4, 5, 6mapsnf1o2 8934 . . . . . . . 8 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)):(𝐵m 1o)–1-1-onto𝐵
10 f1ococnv2 6875 . . . . . . . 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, 11eqtr3id 2791 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
1312coeq2d 5873 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
141, 13eqtrid 2789 . . . 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 22354 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
18 fcoi1 6782 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
1915, 17, 183syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2014, 19eqtrd 2777 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = 𝐴)
21 pf1ind.cp . . . 4 + = (+g𝑅)
22 pf1ind.ct . . . 4 · = (.r𝑅)
23 eqid 2737 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
2423, 3evlval 22119 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
2524rneqi 5948 . . . 4 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26 an4 656 . . . . . 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 2737 . . . . . . . . . . . 12 ran (1o eval 𝑅) = ran (1o eval 𝑅)
2816, 3, 27mpfpf1 22355 . . . . . . . . . . 11 (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
2916, 3, 27mpfpf1 22355 . . . . . . . . . . 11 (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30 vex 3484 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
31 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3230, 31elab 3679 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
33 eleq1 2829 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3432, 33bitr3id 285 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3534anbi1d 631 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3635anbi1d 631 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
37 ovex 7464 . . . . . . . . . . . . . . 15 (𝑓f + 𝑔) ∈ V
38 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))
3937, 38elab 3679 . . . . . . . . . . . . . 14 ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
40 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
4140eleq1d 2826 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥𝜓}))
4239, 41bitr3id 285 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜁 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥𝜓}))
4336, 42imbi12d 344 . . . . . . . . . . . 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 3484 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
45 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4644, 45elab 3679 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
47 eleq1 2829 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4846, 47bitr3id 285 . . . . . . . . . . . . . . 15 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4948anbi2d 630 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})))
5049anbi1d 631 . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
5251eleq1d 2826 . . . . . . . . . . . . 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 344 . . . . . . . . . . . 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 413 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5655an4s 660 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5756expimpd 453 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5843, 53, 57vtocl2ga 3578 . . . . . . . . . . 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 596 . . . . . . . . . 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 416 . . . . . . . . 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 407 . . . . . . . 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 22128 . . . . . . . . . . . 12 (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵m 1o)⟶𝐵)
6362ad2antrl 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵m 1o)⟶𝐵)
6463ffnd 6737 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵m 1o))
6525, 3mpff 22128 . . . . . . . . . . . 12 (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵m 1o)⟶𝐵)
6665ad2antll 729 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵m 1o)⟶𝐵)
6766ffnd 6737 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵m 1o))
68 eqid 2737 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1o × {𝑤})) = (𝑤𝐵 ↦ (1o × {𝑤}))
692, 4, 5, 68mapsnf1o3 8935 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1o × {𝑤})):𝐵1-1-onto→(𝐵m 1o)
70 f1of 6848 . . . . . . . . . . 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 7466 . . . . . . . . . 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 4227 . . . . . . . . . 10 ((𝐵m 1o) ∩ (𝐵m 1o)) = (𝐵m 1o)
7564, 67, 71, 72, 72, 73, 74ofco 7722 . . . . . . . . 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 2826 . . . . . . . 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 259 . . . . . . 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 453 . . . . . 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, 78biimtrid 242 . . . . 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 406 . . . 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 7464 . . . . . . . . . . . . . . 15 (𝑓f · 𝑔) ∈ V
82 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))
8381, 82elab 3679 . . . . . . . . . . . . . 14 ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
84 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
8584eleq1d 2826 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥𝜓}))
8683, 85bitr3id 285 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜎 ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥𝜓}))
8736, 86imbi12d 344 . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
8988eleq1d 2826 . . . . . . . . . . . . 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 344 . . . . . . . . . . . 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 413 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9392an4s 660 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9493expimpd 453 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9587, 90, 94vtocl2ga 3578 . . . . . . . . . . 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 596 . . . . . . . . . 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 416 . . . . . . . . 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 407 . . . . . . . 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 7722 . . . . . . . . 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 2826 . . . . . . . 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 259 . . . . . . 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 453 . . . . . 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, 102biimtrid 242 . . . . 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 406 . . . 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 5868 . . . . 5 (𝑦 = ((𝐵m 1o) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
106105eleq1d 2826 . . . 4 (𝑦 = ((𝐵m 1o) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
107 coeq1 5868 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
108107eleq1d 2826 . . . 4 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
109 coeq1 5868 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
110109eleq1d 2826 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5868 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
112111eleq1d 2826 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5868 . . . . 5 (𝑦 = (𝑎f + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
114113eleq1d 2826 . . . 4 (𝑦 = (𝑎f + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5868 . . . . 5 (𝑦 = (𝑎f · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
116115eleq1d 2826 . . . 4 (𝑦 = (𝑎f · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5868 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
118117eleq1d 2826 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
11916pf1rcl 22353 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12015, 119syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
121120adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
122 1on 8518 . . . . . . . . . . . 12 1o ∈ On
123 eqid 2737 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
124123mplassa 22042 . . . . . . . . . . . 12 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125122, 120, 124sylancr 587 . . . . . . . . . . 11 (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126 eqid 2737 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
127 eqid 2737 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
128126, 127ply1ascl 22261 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1o mPoly 𝑅))
129 eqid 2737 . . . . . . . . . . . 12 (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130128, 129asclrhm 21910 . . . . . . . . . . 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 22033 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1o mPoly 𝑅)))
134133oveq1d 7446 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135131, 134eleqtrrd 2844 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136 eqid 2737 . . . . . . . . . 10 (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
1373, 136rhmf 20485 . . . . . . . . 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)))
139138ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140 eqid 2737 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
141140, 23, 3, 123, 136evl1val 22333 . . . . . . 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 22338 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144120, 143sylan 580 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1453ressid 17290 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
146121, 145syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
147146oveq2d 7447 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1o mPoly (𝑅s 𝐵)) = (1o mPoly 𝑅))
148147fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149148, 128eqtr4di 2795 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
150149fveq1d 6908 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
151150fveq2d 6910 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
152 eqid 2737 . . . . . . . . 9 (1o mPoly (𝑅s 𝐵)) = (1o mPoly (𝑅s 𝐵))
153 eqid 2737 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
154 eqid 2737 . . . . . . . . 9 (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly (𝑅s 𝐵)))
155122a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1o ∈ On)
156 crngring 20242 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1573subrgid 20573 . . . . . . . . . . 11 (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
158120, 156, 1573syl 18 . . . . . . . . . 10 (𝜑𝐵 ∈ (SubRing‘𝑅))
159158adantr 480 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝐵 ∈ (SubRing‘𝑅))
160 simpr 484 . . . . . . . . 9 ((𝜑𝑎𝐵) → 𝑎𝐵)
16124, 152, 153, 3, 154, 155, 121, 159, 160evlssca 22113 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
162151, 161eqtr3d 2779 . . . . . . 7 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
163162coeq1d 5872 . . . . . 6 ((𝜑𝑎𝐵) → (((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
164142, 144, 1633eqtr3d 2785 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
165 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
166 vsnex 5434 . . . . . . . . . 10 {𝑓} ∈ V
1674, 166xpex 7773 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
168 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
169167, 168elab 3679 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
170165, 169sylibr 234 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
171170ralrimiva 3146 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
172 sneq 4636 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
173172xpeq2d 5715 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174173eleq1d 2826 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
175174rspccva 3621 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
176171, 175sylan 580 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
177164, 176eqeltrrd 2842 . . . 4 ((𝜑𝑎𝐵) → (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
178 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
179 resiexg 7934 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1804, 179ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
181 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
182180, 181elab 3679 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
183178, 182sylibr 234 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18412, 183eqeltrd 2841 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
185 el1o 8533 . . . . . . . . . 10 (𝑎 ∈ 1o𝑎 = ∅)
186 fveq2 6906 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
187185, 186sylbi 217 . . . . . . . . 9 (𝑎 ∈ 1o → (𝑏𝑎) = (𝑏‘∅))
188187mpteq2dv 5244 . . . . . . . 8 (𝑎 ∈ 1o → (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)))
189188coeq1d 5872 . . . . . . 7 (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
190189eleq1d 2826 . . . . . 6 (𝑎 ∈ 1o → (((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
191184, 190syl5ibrcom 247 . . . . 5 (𝜑 → (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
192191imp 406 . . . 4 ((𝜑𝑎 ∈ 1o) → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
19316, 3, 27pf1mpf 22356 . . . . 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 22131 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
19620, 195eqeltrrd 2842 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
197 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
198197elabg 3676 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
19915, 198syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
200196, 199mpbid 232 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  c0 4333  {csn 4626  cmpt 5225   I cid 5577   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  ccom 5689  Oncon0 6384  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  f cof 7695  1oc1o 8499  m cmap 8866  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469  SubRingcsubrg 20569  AssAlgcasa 21870  algSccascl 21872   mPoly cmpl 21926   evalSub ces 22096   eval cevl 22097  Poly1cpl1 22178  eval1ce1 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-lsp 20970  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-ply1 22183  df-evl1 22320
This theorem is referenced by:  pl1cn  33954
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