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Theorem pf1ind 21721
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 6217 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
2 df1o2 8419 . . . . . . . . 9 1o = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
43fvexi 6856 . . . . . . . . 9 𝐵 ∈ V
5 0ex 5264 . . . . . . . . 9 ∅ ∈ V
6 eqid 2736 . . . . . . . . 9 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))
72, 4, 5, 6mapsncnv 8831 . . . . . . . 8 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1o × {𝑤}))
87coeq2i 5816 . . . . . . 7 ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))
92, 4, 5, 6mapsnf1o2 8832 . . . . . . . 8 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)):(𝐵m 1o)–1-1-onto𝐵
10 f1ococnv2 6811 . . . . . . . 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 2790 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
1312coeq2d 5818 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
141, 13eqtrid 2788 . . . 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 21716 . . . . 5 (𝐴𝑄𝐴:𝐵𝐵)
18 fcoi1 6716 . . . . 5 (𝐴:𝐵𝐵 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
1915, 17, 183syl 18 . . . 4 (𝜑 → (𝐴 ∘ ( I ↾ 𝐵)) = 𝐴)
2014, 19eqtrd 2776 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = 𝐴)
21 pf1ind.cp . . . 4 + = (+g𝑅)
22 pf1ind.ct . . . 4 · = (.r𝑅)
23 eqid 2736 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
2423, 3evlval 21505 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
2524rneqi 5892 . . . 4 ran (1o eval 𝑅) = ran ((1o evalSub 𝑅)‘𝐵)
26 an4 654 . . . . . 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 2736 . . . . . . . . . . . 12 ran (1o eval 𝑅) = ran (1o eval 𝑅)
2816, 3, 27mpfpf1 21717 . . . . . . . . . . 11 (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
2916, 3, 27mpfpf1 21717 . . . . . . . . . . 11 (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30 vex 3449 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
31 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3230, 31elab 3630 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
33 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓 ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3432, 33bitr3id 284 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜏 ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
3534anbi1d 630 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝜏𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂)))
3635anbi1d 630 . . . . . . . . . . . . 13 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝜏𝜂) ∧ 𝜑) ↔ (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ∧ 𝜑)))
37 ovex 7390 . . . . . . . . . . . . . . 15 (𝑓f + 𝑔) ∈ V
38 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))
3937, 38elab 3630 . . . . . . . . . . . . . 14 ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
40 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
4140eleq1d 2822 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) ∈ {𝑥𝜓}))
4239, 41bitr3id 284 . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
45 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4644, 45elab 3630 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
47 eleq1 2825 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑔 ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4846, 47bitr3id 284 . . . . . . . . . . . . . . 15 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝜂 ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
4948anbi2d 629 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ 𝜂) ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ∧ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})))
5049anbi1d 630 . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
5251eleq1d 2822 . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜁))
5655an4s 658 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜁))
5756expimpd 454 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜁))
5843, 53, 57vtocl2ga 3535 . . . . . . . . . . 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 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 21514 . . . . . . . . . . . 12 (𝑎 ∈ ran (1o eval 𝑅) → 𝑎:(𝐵m 1o)⟶𝐵)
6362ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎:(𝐵m 1o)⟶𝐵)
6463ffnd 6669 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑎 Fn (𝐵m 1o))
6525, 3mpff 21514 . . . . . . . . . . . 12 (𝑏 ∈ ran (1o eval 𝑅) → 𝑏:(𝐵m 1o)⟶𝐵)
6665ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏:(𝐵m 1o)⟶𝐵)
6766ffnd 6669 . . . . . . . . . 10 ((𝜑 ∧ (𝑎 ∈ ran (1o eval 𝑅) ∧ 𝑏 ∈ ran (1o eval 𝑅))) → 𝑏 Fn (𝐵m 1o))
68 eqid 2736 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1o × {𝑤})) = (𝑤𝐵 ↦ (1o × {𝑤}))
692, 4, 5, 68mapsnf1o3 8833 . . . . . . . . . . 11 (𝑤𝐵 ↦ (1o × {𝑤})):𝐵1-1-onto→(𝐵m 1o)
70 f1of 6784 . . . . . . . . . . 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 7392 . . . . . . . . . 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 4178 . . . . . . . . . 10 ((𝐵m 1o) ∩ (𝐵m 1o)) = (𝐵m 1o)
7564, 67, 71, 72, 72, 73, 74ofco 7640 . . . . . . . . 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 2822 . . . . . . . 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 258 . . . . . . 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, 78biimtrid 241 . . . . 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 7390 . . . . . . . . . . . . . . 15 (𝑓f · 𝑔) ∈ V
82 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))
8381, 82elab 3630 . . . . . . . . . . . . . 14 ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
84 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
8584eleq1d 2822 . . . . . . . . . . . . . 14 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) ∈ {𝑥𝜓}))
8683, 85bitr3id 284 . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
8988eleq1d 2822 . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 14 (((𝑓𝑄𝜏) ∧ (𝑔𝑄𝜂)) → (𝜑𝜎))
9392an4s 658 . . . . . . . . . . . . 13 (((𝑓𝑄𝑔𝑄) ∧ (𝜏𝜂)) → (𝜑𝜎))
9493expimpd 454 . . . . . . . . . . . 12 ((𝑓𝑄𝑔𝑄) → (((𝜏𝜂) ∧ 𝜑) → 𝜎))
9587, 90, 94vtocl2ga 3535 . . . . . . . . . . 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 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 7640 . . . . . . . . 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 2822 . . . . . . . 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 258 . . . . . . 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, 102biimtrid 241 . . . . 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 5813 . . . . 5 (𝑦 = ((𝐵m 1o) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
106105eleq1d 2822 . . . 4 (𝑦 = ((𝐵m 1o) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
107 coeq1 5813 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
108107eleq1d 2822 . . . 4 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
109 coeq1 5813 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
110109eleq1d 2822 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5813 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
112111eleq1d 2822 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5813 . . . . 5 (𝑦 = (𝑎f + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
114113eleq1d 2822 . . . 4 (𝑦 = (𝑎f + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5813 . . . . 5 (𝑦 = (𝑎f · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
116115eleq1d 2822 . . . 4 (𝑦 = (𝑎f · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5813 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
118117eleq1d 2822 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
11916pf1rcl 21715 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12015, 119syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
121120adantr 481 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
122 1on 8424 . . . . . . . . . . . 12 1o ∈ On
123 eqid 2736 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
124123mplassa 21427 . . . . . . . . . . . 12 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125122, 120, 124sylancr 587 . . . . . . . . . . 11 (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126 eqid 2736 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
127 eqid 2736 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
128126, 127ply1ascl 21629 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1o mPoly 𝑅))
129 eqid 2736 . . . . . . . . . . . 12 (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130128, 129asclrhm 21293 . . . . . . . . . . 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 21417 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1o mPoly 𝑅)))
134133oveq1d 7372 . . . . . . . . . 10 (𝜑 → (𝑅 RingHom (1o mPoly 𝑅)) = ((Scalar‘(1o mPoly 𝑅)) RingHom (1o mPoly 𝑅)))
135131, 134eleqtrrd 2841 . . . . . . . . 9 (𝜑 → (algSc‘(Poly1𝑅)) ∈ (𝑅 RingHom (1o mPoly 𝑅)))
136 eqid 2736 . . . . . . . . . 10 (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
1373, 136rhmf 20158 . . . . . . . . 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 7035 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140 eqid 2736 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
141140, 23, 3, 123, 136evl1val 21695 . . . . . . 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 21700 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144120, 143sylan 580 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1453ressid 17125 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
146121, 145syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
147146oveq2d 7373 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1o mPoly (𝑅s 𝐵)) = (1o mPoly 𝑅))
148147fveq2d 6846 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149148, 128eqtr4di 2794 . . . . . . . . . 10 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(Poly1𝑅)))
150149fveq1d 6844 . . . . . . . . 9 ((𝜑𝑎𝐵) → ((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎) = ((algSc‘(Poly1𝑅))‘𝑎))
151150fveq2d 6846 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)))
152 eqid 2736 . . . . . . . . 9 (1o mPoly (𝑅s 𝐵)) = (1o mPoly (𝑅s 𝐵))
153 eqid 2736 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
154 eqid 2736 . . . . . . . . 9 (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly (𝑅s 𝐵)))
155122a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1o ∈ On)
156 crngring 19976 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1573subrgid 20224 . . . . . . . . . . 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 21499 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
162151, 161eqtr3d 2778 . . . . . . 7 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
163162coeq1d 5817 . . . . . 6 ((𝜑𝑎𝐵) → (((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
164142, 144, 1633eqtr3d 2784 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
165 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
166 vsnex 5386 . . . . . . . . . 10 {𝑓} ∈ V
1674, 166xpex 7687 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
168 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
169167, 168elab 3630 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
170165, 169sylibr 233 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
171170ralrimiva 3143 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
172 sneq 4596 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
173172xpeq2d 5663 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174173eleq1d 2822 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
175174rspccva 3580 . . . . . 6 ((∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓} ∧ 𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
176171, 175sylan 580 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) ∈ {𝑥𝜓})
177164, 176eqeltrrd 2839 . . . 4 ((𝜑𝑎𝐵) → (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
178 pf1ind.pr . . . . . . . 8 (𝜑𝜃)
179 resiexg 7851 . . . . . . . . . 10 (𝐵 ∈ V → ( I ↾ 𝐵) ∈ V)
1804, 179ax-mp 5 . . . . . . . . 9 ( I ↾ 𝐵) ∈ V
181 pf1ind.wb . . . . . . . . 9 (𝑥 = ( I ↾ 𝐵) → (𝜓𝜃))
182180, 181elab 3630 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
183178, 182sylibr 233 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18412, 183eqeltrd 2838 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
185 el1o 8441 . . . . . . . . . 10 (𝑎 ∈ 1o𝑎 = ∅)
186 fveq2 6842 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
187185, 186sylbi 216 . . . . . . . . 9 (𝑎 ∈ 1o → (𝑏𝑎) = (𝑏‘∅))
188187mpteq2dv 5207 . . . . . . . 8 (𝑎 ∈ 1o → (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)))
189188coeq1d 5817 . . . . . . 7 (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
190189eleq1d 2822 . . . . . 6 (𝑎 ∈ 1o → (((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
191184, 190syl5ibrcom 246 . . . . 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 21718 . . . . 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 21517 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
19620, 195eqeltrrd 2839 . 2 (𝜑𝐴 ∈ {𝑥𝜓})
197 pf1ind.wg . . . 4 (𝑥 = 𝐴 → (𝜓𝜌))
198197elabg 3628 . . 3 (𝐴𝑄 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
19915, 198syl 17 . 2 (𝜑 → (𝐴 ∈ {𝑥𝜓} ↔ 𝜌))
200196, 199mpbid 231 1 (𝜑𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  wral 3064  Vcvv 3445  c0 4282  {csn 4586  cmpt 5188   I cid 5530   × cxp 5631  ccnv 5632  ran crn 5634  cres 5635  ccom 5637  Oncon0 6317  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  f cof 7615  1oc1o 8405  m cmap 8765  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  Scalarcsca 17136  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  SubRingcsubrg 20218  AssAlgcasa 21256  algSccascl 21258   mPoly cmpl 21308   evalSub ces 21480   eval cevl 21481  Poly1cpl1 21548  eval1ce1 21680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-evls 21482  df-evl 21483  df-psr1 21551  df-ply1 21553  df-evl1 21682
This theorem is referenced by:  pl1cn  32536
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