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Theorem pf1ind 22374
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 6286 . . . . 5 ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
2 df1o2 8511 . . . . . . . . 9 1o = {∅}
3 pf1ind.cb . . . . . . . . . 10 𝐵 = (Base‘𝑅)
43fvexi 6920 . . . . . . . . 9 𝐵 ∈ V
5 0ex 5312 . . . . . . . . 9 ∅ ∈ V
6 eqid 2734 . . . . . . . . 9 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))
72, 4, 5, 6mapsncnv 8931 . . . . . . . 8 (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) = (𝑤𝐵 ↦ (1o × {𝑤}))
87coeq2i 5873 . . . . . . 7 ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))
92, 4, 5, 6mapsnf1o2 8932 . . . . . . . 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 2788 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ( I ↾ 𝐵))
1312coeq2d 5875 . . . . 5 (𝜑 → (𝐴 ∘ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤})))) = (𝐴 ∘ ( I ↾ 𝐵)))
141, 13eqtrid 2786 . . . 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 22369 . . . . 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 2774 . . 3 (𝜑 → ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = 𝐴)
21 pf1ind.cp . . . 4 + = (+g𝑅)
22 pf1ind.ct . . . 4 · = (.r𝑅)
23 eqid 2734 . . . . . 6 (1o eval 𝑅) = (1o eval 𝑅)
2423, 3evlval 22136 . . . . 5 (1o eval 𝑅) = ((1o evalSub 𝑅)‘𝐵)
2524rneqi 5950 . . . 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 2734 . . . . . . . . . . . 12 ran (1o eval 𝑅) = ran (1o eval 𝑅)
2816, 3, 27mpfpf1 22370 . . . . . . . . . . 11 (𝑎 ∈ ran (1o eval 𝑅) → (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
2916, 3, 27mpfpf1 22370 . . . . . . . . . . 11 (𝑏 ∈ ran (1o eval 𝑅) → (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ 𝑄)
30 vex 3481 . . . . . . . . . . . . . . . . 17 𝑓 ∈ V
31 pf1ind.wc . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑓 → (𝜓𝜏))
3230, 31elab 3680 . . . . . . . . . . . . . . . 16 (𝑓 ∈ {𝑥𝜓} ↔ 𝜏)
33 eleq1 2826 . . . . . . . . . . . . . . . 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 7463 . . . . . . . . . . . . . . 15 (𝑓f + 𝑔) ∈ V
38 pf1ind.we . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f + 𝑔) → (𝜓𝜁))
3937, 38elab 3680 . . . . . . . . . . . . . 14 ((𝑓f + 𝑔) ∈ {𝑥𝜓} ↔ 𝜁)
40 oveq1 7437 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔))
4140eleq1d 2823 . . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . . . 17 𝑔 ∈ V
45 pf1ind.wd . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑔 → (𝜓𝜂))
4644, 45elab 3680 . . . . . . . . . . . . . . . 16 (𝑔 ∈ {𝑥𝜓} ↔ 𝜂)
47 eleq1 2826 . . . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f + (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
5251eleq1d 2823 . . . . . . . . . . . . 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 3577 . . . . . . . . . . 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 22145 . . . . . . . . . . . 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 22145 . . . . . . . . . . . 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 2734 . . . . . . . . . . . 12 (𝑤𝐵 ↦ (1o × {𝑤})) = (𝑤𝐵 ↦ (1o × {𝑤}))
692, 4, 5, 68mapsnf1o3 8933 . . . . . . . . . . 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 7465 . . . . . . . . . 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 4234 . . . . . . . . . 10 ((𝐵m 1o) ∩ (𝐵m 1o)) = (𝐵m 1o)
7564, 67, 71, 72, 72, 73, 74ofco 7721 . . . . . . . . 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 2823 . . . . . . . 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 7463 . . . . . . . . . . . . . . 15 (𝑓f · 𝑔) ∈ V
82 pf1ind.wf . . . . . . . . . . . . . . 15 (𝑥 = (𝑓f · 𝑔) → (𝜓𝜎))
8381, 82elab 3680 . . . . . . . . . . . . . 14 ((𝑓f · 𝑔) ∈ {𝑥𝜓} ↔ 𝜎)
84 oveq1 7437 . . . . . . . . . . . . . . 15 (𝑓 = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → (𝑓f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔))
8584eleq1d 2823 . . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . 14 (𝑔 = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) → ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · 𝑔) = ((𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∘f · (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤})))))
8988eleq1d 2823 . . . . . . . . . . . . 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 3577 . . . . . . . . . . 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 7721 . . . . . . . . 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 2823 . . . . . . . 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 5870 . . . . 5 (𝑦 = ((𝐵m 1o) × {𝑎}) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
106105eleq1d 2823 . . . 4 (𝑦 = ((𝐵m 1o) × {𝑎}) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
107 coeq1 5870 . . . . 5 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
108107eleq1d 2823 . . . 4 (𝑦 = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
109 coeq1 5870 . . . . 5 (𝑦 = 𝑎 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
110109eleq1d 2823 . . . 4 (𝑦 = 𝑎 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑎 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
111 coeq1 5870 . . . . 5 (𝑦 = 𝑏 → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
112111eleq1d 2823 . . . 4 (𝑦 = 𝑏 → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ (𝑏 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
113 coeq1 5870 . . . . 5 (𝑦 = (𝑎f + 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
114113eleq1d 2823 . . . 4 (𝑦 = (𝑎f + 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f + 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
115 coeq1 5870 . . . . 5 (𝑦 = (𝑎f · 𝑏) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
116115eleq1d 2823 . . . 4 (𝑦 = (𝑎f · 𝑏) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝑎f · 𝑏) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
117 coeq1 5870 . . . . 5 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → (𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
118117eleq1d 2823 . . . 4 (𝑦 = (𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) → ((𝑦 ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓} ↔ ((𝐴 ∘ (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅))) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓}))
11916pf1rcl 22368 . . . . . . . . 9 (𝐴𝑄𝑅 ∈ CRing)
12015, 119syl 17 . . . . . . . 8 (𝜑𝑅 ∈ CRing)
121120adantr 480 . . . . . . 7 ((𝜑𝑎𝐵) → 𝑅 ∈ CRing)
122 1on 8516 . . . . . . . . . . . 12 1o ∈ On
123 eqid 2734 . . . . . . . . . . . . 13 (1o mPoly 𝑅) = (1o mPoly 𝑅)
124123mplassa 22059 . . . . . . . . . . . 12 ((1o ∈ On ∧ 𝑅 ∈ CRing) → (1o mPoly 𝑅) ∈ AssAlg)
125122, 120, 124sylancr 587 . . . . . . . . . . 11 (𝜑 → (1o mPoly 𝑅) ∈ AssAlg)
126 eqid 2734 . . . . . . . . . . . . 13 (Poly1𝑅) = (Poly1𝑅)
127 eqid 2734 . . . . . . . . . . . . 13 (algSc‘(Poly1𝑅)) = (algSc‘(Poly1𝑅))
128126, 127ply1ascl 22276 . . . . . . . . . . . 12 (algSc‘(Poly1𝑅)) = (algSc‘(1o mPoly 𝑅))
129 eqid 2734 . . . . . . . . . . . 12 (Scalar‘(1o mPoly 𝑅)) = (Scalar‘(1o mPoly 𝑅))
130128, 129asclrhm 21927 . . . . . . . . . . 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 22050 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘(1o mPoly 𝑅)))
134133oveq1d 7445 . . . . . . . . . 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 2734 . . . . . . . . . 10 (Base‘(1o mPoly 𝑅)) = (Base‘(1o mPoly 𝑅))
1373, 136rhmf 20501 . . . . . . . . 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 7103 . . . . . . 7 ((𝜑𝑎𝐵) → ((algSc‘(Poly1𝑅))‘𝑎) ∈ (Base‘(1o mPoly 𝑅)))
140 eqid 2734 . . . . . . . 8 (eval1𝑅) = (eval1𝑅)
141140, 23, 3, 123, 136evl1val 22348 . . . . . . 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 22353 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
144120, 143sylan 580 . . . . . 6 ((𝜑𝑎𝐵) → ((eval1𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = (𝐵 × {𝑎}))
1453ressid 17289 . . . . . . . . . . . . . 14 (𝑅 ∈ CRing → (𝑅s 𝐵) = 𝑅)
146121, 145syl 17 . . . . . . . . . . . . 13 ((𝜑𝑎𝐵) → (𝑅s 𝐵) = 𝑅)
147146oveq2d 7446 . . . . . . . . . . . 12 ((𝜑𝑎𝐵) → (1o mPoly (𝑅s 𝐵)) = (1o mPoly 𝑅))
148147fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑎𝐵) → (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly 𝑅)))
149148, 128eqtr4di 2792 . . . . . . . . . 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 2734 . . . . . . . . 9 (1o mPoly (𝑅s 𝐵)) = (1o mPoly (𝑅s 𝐵))
153 eqid 2734 . . . . . . . . 9 (𝑅s 𝐵) = (𝑅s 𝐵)
154 eqid 2734 . . . . . . . . 9 (algSc‘(1o mPoly (𝑅s 𝐵))) = (algSc‘(1o mPoly (𝑅s 𝐵)))
155122a1i 11 . . . . . . . . 9 ((𝜑𝑎𝐵) → 1o ∈ On)
156 crngring 20262 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1573subrgid 20589 . . . . . . . . . . 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 22130 . . . . . . . 8 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(1o mPoly (𝑅s 𝐵)))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
162151, 161eqtr3d 2776 . . . . . . 7 ((𝜑𝑎𝐵) → ((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) = ((𝐵m 1o) × {𝑎}))
163162coeq1d 5874 . . . . . 6 ((𝜑𝑎𝐵) → (((1o eval 𝑅)‘((algSc‘(Poly1𝑅))‘𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
164142, 144, 1633eqtr3d 2782 . . . . 5 ((𝜑𝑎𝐵) → (𝐵 × {𝑎}) = (((𝐵m 1o) × {𝑎}) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
165 pf1ind.co . . . . . . . 8 ((𝜑𝑓𝐵) → 𝜒)
166 vsnex 5439 . . . . . . . . . 10 {𝑓} ∈ V
1674, 166xpex 7771 . . . . . . . . 9 (𝐵 × {𝑓}) ∈ V
168 pf1ind.wa . . . . . . . . 9 (𝑥 = (𝐵 × {𝑓}) → (𝜓𝜒))
169167, 168elab 3680 . . . . . . . 8 ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ 𝜒)
170165, 169sylibr 234 . . . . . . 7 ((𝜑𝑓𝐵) → (𝐵 × {𝑓}) ∈ {𝑥𝜓})
171170ralrimiva 3143 . . . . . 6 (𝜑 → ∀𝑓𝐵 (𝐵 × {𝑓}) ∈ {𝑥𝜓})
172 sneq 4640 . . . . . . . . 9 (𝑓 = 𝑎 → {𝑓} = {𝑎})
173172xpeq2d 5718 . . . . . . . 8 (𝑓 = 𝑎 → (𝐵 × {𝑓}) = (𝐵 × {𝑎}))
174173eleq1d 2823 . . . . . . 7 (𝑓 = 𝑎 → ((𝐵 × {𝑓}) ∈ {𝑥𝜓} ↔ (𝐵 × {𝑎}) ∈ {𝑥𝜓}))
175174rspccva 3620 . . . . . 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 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 3680 . . . . . . . 8 (( I ↾ 𝐵) ∈ {𝑥𝜓} ↔ 𝜃)
183178, 182sylibr 234 . . . . . . 7 (𝜑 → ( I ↾ 𝐵) ∈ {𝑥𝜓})
18412, 183eqeltrd 2838 . . . . . 6 (𝜑 → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) ∈ {𝑥𝜓})
185 el1o 8531 . . . . . . . . . 10 (𝑎 ∈ 1o𝑎 = ∅)
186 fveq2 6906 . . . . . . . . . 10 (𝑎 = ∅ → (𝑏𝑎) = (𝑏‘∅))
187185, 186sylbi 217 . . . . . . . . 9 (𝑎 ∈ 1o → (𝑏𝑎) = (𝑏‘∅))
188187mpteq2dv 5249 . . . . . . . 8 (𝑎 ∈ 1o → (𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) = (𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)))
189188coeq1d 5874 . . . . . . 7 (𝑎 ∈ 1o → ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏𝑎)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))) = ((𝑏 ∈ (𝐵m 1o) ↦ (𝑏‘∅)) ∘ (𝑤𝐵 ↦ (1o × {𝑤}))))
190189eleq1d 2823 . . . . . 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 22371 . . . . 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 22148 . . 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 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 1536  wcel 2105  {cab 2711  wral 3058  Vcvv 3477  c0 4338  {csn 4630  cmpt 5230   I cid 5581   × cxp 5686  ccnv 5687  ran crn 5689  cres 5690  ccom 5692  Oncon0 6385  wf 6558  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  f cof 7694  1oc1o 8497  m cmap 8864  Basecbs 17244  s cress 17273  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300  Ringcrg 20250  CRingccrg 20251   RingHom crh 20485  SubRingcsubrg 20585  AssAlgcasa 21887  algSccascl 21889   mPoly cmpl 21943   evalSub ces 22113   eval cevl 22114  Poly1cpl1 22193  eval1ce1 22333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-lsp 20987  df-assa 21890  df-asp 21891  df-ascl 21892  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-evls 22115  df-evl 22116  df-psr1 22196  df-ply1 22198  df-evl1 22335
This theorem is referenced by:  pl1cn  33915
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