Theorem constrextdg2lem 34047

Description: Lemma for constrextdg2 34048. (Contributed by Thierry Arnoux, 19-Oct-2025.)

Hypotheses

Ref	Expression
constr0.1	⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
constrextdg2.1	⊢ 𝐸 = (ℂfld ↾s 𝑒)
constrextdg2.2	⊢ 𝐹 = (ℂfld ↾s 𝑓)
constrextdg2.l	⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
constrextdg2.n	⊢ (𝜑 → 𝑁 ∈ ω)
constrextdg2lem.1	⊢ (𝜑 → 𝑅 ∈ ( < Chain (SubDRing‘ℂfld)))
constrextdg2lem.2	⊢ (𝜑 → (𝑅‘0) = ℚ)
constrextdg2lem.3	⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))

Assertion

Ref	Expression
constrextdg2lem	⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))

Distinct variable groups:   < ,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥   𝑁,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥   𝑣,𝑅   𝜑,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓,𝑟,𝑠,𝑡,𝑣,𝑥

Allowed substitution hints:   𝑅(𝑥,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐸(𝑥,𝑣,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)   𝐹(𝑥,𝑣,𝑡,𝑒,𝑓,𝑠,𝑟,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem constrextdg2lem

Dummy variables 𝑔 𝑦 𝑖 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq2 4117	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ ∅))
2	1	sseq1d 3969	. . . . 5 ⊢ (𝑖 = ∅ → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
3	2	anbi2d 639	. . . 4 ⊢ (𝑖 = ∅ → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣))))
4	3	rexbidv 3188	. . 3 ⊢ (𝑖 = ∅ → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣))))
5		uneq2 4117	. . . . . 6 ⊢ (𝑖 = 𝑔 → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ 𝑔))
6	5	sseq1d 3969	. . . . 5 ⊢ (𝑖 = 𝑔 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)))
7	6	anbi2d 639	. . . 4 ⊢ (𝑖 = 𝑔 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))))
8	7	rexbidv 3188	. . 3 ⊢ (𝑖 = 𝑔 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))))
9		fveq1 6868	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
10	9	eqeq1d 2766	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
11		fveq2 6869	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
12	11	sseq2d 3970	. . . . . 6 ⊢ (𝑣 = 𝑢 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)))
13	10, 12	anbi12d 641	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢))))
14	13	cbvrexvw 3243	. . . 4 ⊢ (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)))
15		uneq2 4117	. . . . . . 7 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})))
16	15	sseq1d 3969	. . . . . 6 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
17	16	anbi2d 639	. . . . 5 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
18	17	rexbidv 3188	. . . 4 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
19	14, 18	bitrid 285	. . 3 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
20		uneq2 4117	. . . . . 6 ⊢ (𝑖 = (𝐶‘suc 𝑁) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)))
21	20	sseq1d 3969	. . . . 5 ⊢ (𝑖 = (𝐶‘suc 𝑁) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)))
22	21	anbi2d 639	. . . 4 ⊢ (𝑖 = (𝐶‘suc 𝑁) → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣))))
23	22	rexbidv 3188	. . 3 ⊢ (𝑖 = (𝐶‘suc 𝑁) → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣))))
24		fveq1 6868	. . . . . 6 ⊢ (𝑣 = 𝑅 → (𝑣‘0) = (𝑅‘0))
25	24	eqeq1d 2766	. . . . 5 ⊢ (𝑣 = 𝑅 → ((𝑣‘0) = ℚ ↔ (𝑅‘0) = ℚ))
26		fveq2 6869	. . . . . 6 ⊢ (𝑣 = 𝑅 → (lastS‘𝑣) = (lastS‘𝑅))
27	26	sseq2d 3970	. . . . 5 ⊢ (𝑣 = 𝑅 → (((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅)))
28	25, 27	anbi12d 641	. . . 4 ⊢ (𝑣 = 𝑅 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)) ↔ ((𝑅‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅))))
29		constrextdg2lem.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ( < Chain (SubDRing‘ℂfld)))
30		constrextdg2lem.2	. . . . 5 ⊢ (𝜑 → (𝑅‘0) = ℚ)
31		un0 4350	. . . . . 6 ⊢ ((𝐶‘𝑁) ∪ ∅) = (𝐶‘𝑁)
32		constrextdg2lem.3	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))
33	31, 32	eqsstrid 3976	. . . . 5 ⊢ (𝜑 → ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅))
34	30, 33	jca 519	. . . 4 ⊢ (𝜑 → ((𝑅‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅)))
35	28, 29, 34	rspcedvdw 3586	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
36		fveq1 6868	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢‘0) = (𝑣‘0))
37	36	eqeq1d 2766	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢‘0) = ℚ ↔ (𝑣‘0) = ℚ))
38		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (lastS‘𝑢) = (lastS‘𝑣))
39	38	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣)))
40	37, 39	anbi12d 641	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣))))
41		simpllr 785	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
42	41	adantr 484	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
43		simpllr 785	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → (𝑣‘0) = ℚ)
44		simpr 488	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))
45	44	unssad 4147	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → (𝐶‘𝑁) ⊆ (lastS‘𝑣))
46	45	adantr 484	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → (𝐶‘𝑁) ⊆ (lastS‘𝑣))
47		simplr 778	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))
48	47	unssbd 4148	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → 𝑔 ⊆ (lastS‘𝑣))
49		simpr 488	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → 𝑦 ∈ (lastS‘𝑣))
50	49	snssd 4747	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → {𝑦} ⊆ (lastS‘𝑣))
51	48, 50	unssd 4146	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → (𝑔 ∪ {𝑦}) ⊆ (lastS‘𝑣))
52	46, 51	unssd 4146	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣))
53	43, 52	jca 519	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣)))
54	40, 42, 53	rspcedvdw 3586	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ (lastS‘𝑣)) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
55		fveq1 6868	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (𝑢‘0) = ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0))
56	55	eqeq1d 2766	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → ((𝑢‘0) = ℚ ↔ ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ))
57		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (lastS‘𝑢) = (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))
58	57	sseq2d 3970	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩))))
59	56, 58	anbi12d 641	. . . . . . . 8 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)) ↔ (((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))))
60		cnfldbas 21430	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
61		cndrng 21455	. . . . . . . . . . 11 ⊢ ℂfld ∈ DivRing
62	61	a1i 11	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ℂfld ∈ DivRing)
63	41	chnwrd 18642	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ∈ Word (SubDRing‘ℂfld))
64		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → 𝑣 = ∅)
65	64	fveq2d 6873	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = (lastS‘∅))
66		lsw0g 14581	. . . . . . . . . . . . . . . . . . 19 ⊢ (lastS‘∅) = ∅
67	65, 66	eqtrdi 2815	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = ∅)
68		simplr 778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))
69		ssun1 4132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐶‘𝑁) ⊆ ((𝐶‘𝑁) ∪ 𝑔)
70		constr0.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐶 = rec((𝑠 ∈ V ↦ {𝑥 ∈ ℂ ∣ (∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑡 ∈ ℝ ∃𝑟 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ 𝑥 = (𝑐 + (𝑟 · (𝑑 − 𝑐))) ∧ (ℑ‘((∗‘(𝑏 − 𝑎)) · (𝑑 − 𝑐))) ≠ 0) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 ∃𝑡 ∈ ℝ (𝑥 = (𝑎 + (𝑡 · (𝑏 − 𝑎))) ∧ (abs‘(𝑥 − 𝑐)) = (abs‘(𝑒 − 𝑓))) ∨ ∃𝑎 ∈ 𝑠 ∃𝑏 ∈ 𝑠 ∃𝑐 ∈ 𝑠 ∃𝑑 ∈ 𝑠 ∃𝑒 ∈ 𝑠 ∃𝑓 ∈ 𝑠 (𝑎 ≠ 𝑑 ∧ (abs‘(𝑥 − 𝑎)) = (abs‘(𝑏 − 𝑐)) ∧ (abs‘(𝑥 − 𝑑)) = (abs‘(𝑒 − 𝑓))))}), {0, 1})
71		constrextdg2.n	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑁 ∈ ω)
72		nnon 7854	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ On)
74	70, 73	constr01 34041	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
75		c0ex 11175	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
76	75	prnz 4738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {0, 1} ≠ ∅
77		ssn0 4360	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({0, 1} ⊆ (𝐶‘𝑁) ∧ {0, 1} ≠ ∅) → (𝐶‘𝑁) ≠ ∅)
78	74, 76, 77	sylancl 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐶‘𝑁) ≠ ∅)
79		ssn0 4360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶‘𝑁) ⊆ ((𝐶‘𝑁) ∪ 𝑔) ∧ (𝐶‘𝑁) ≠ ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
80	69, 78, 79	sylancr 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
81	80	ad2antrr 736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
82		ssn0 4360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣) ∧ ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅) → (lastS‘𝑣) ≠ ∅)
83	68, 81, 82	syl2anc 593	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) ≠ ∅)
84	83	neneqd 2964	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ¬ (lastS‘𝑣) = ∅)
85	67, 84	pm2.65da 826	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ¬ 𝑣 = ∅)
86	85	neqned 2966	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ≠ ∅)
87	86	ad4antr 742	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) → 𝑣 ≠ ∅)
88	87	an62ds 32656	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ≠ ∅)
89		lswcl 14583	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
90	63, 88, 89	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
91	90	adantr 484	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
92	60	sdrgss 20844	. . . . . . . . . . . 12 ⊢ ((lastS‘𝑣) ∈ (SubDRing‘ℂfld) → (lastS‘𝑣) ⊆ ℂ)
93	91, 92	syl 17	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (lastS‘𝑣) ⊆ ℂ)
94		onsuc 7795	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
95	73, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → suc 𝑁 ∈ On)
96	70, 95	constrsscn 34039	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘suc 𝑁) ⊆ ℂ)
97	96	ad6antr 746	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝐶‘suc 𝑁) ⊆ ℂ)
98		simp-4r 793	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔))
99	98	eldifad 3918	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑦 ∈ (𝐶‘suc 𝑁))
100	99	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑦 ∈ (𝐶‘suc 𝑁))
101	97, 100	sseldd 3939	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑦 ∈ ℂ)
102	101	snssd 4747	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → {𝑦} ⊆ ℂ)
103	93, 102	unssd 4146	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((lastS‘𝑣) ∪ {𝑦}) ⊆ ℂ)
104	60, 62, 103	fldgensdrg 33503	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ∈ (SubDRing‘ℂfld))
105	41	adantr 484	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑣 ∈ ( < Chain (SubDRing‘ℂfld)))
106	91	elexd 3479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (lastS‘𝑣) ∈ V)
107	104	elexd 3479	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ∈ V)
108		eqid 2764	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
109		eqid 2764	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
110		cnfldfld 33530	. . . . . . . . . . . . 13 ⊢ ℂfld ∈ Field
111	110	a1i 11	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ℂfld ∈ Field)
112	60, 108, 109, 111, 91, 102	fldgenfldext 33967	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣)))
113		simpr 488	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2)
114		constrextdg2.1	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (ℂfld ↾s 𝑒)
115		constrextdg2.2	. . . . . . . . . . . . . . . 16 ⊢ 𝐹 = (ℂfld ↾s 𝑓)
116	114, 115	breq12i 5111	. . . . . . . . . . . . . . 15 ⊢ (𝐸/FldExt𝐹 ↔ (ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓))
117		oveq2 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) → (ℂfld ↾s 𝑒) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))))
118	117	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (ℂfld ↾s 𝑒) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))))
119		oveq2 7406	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (lastS‘𝑣) → (ℂfld ↾s 𝑓) = (ℂfld ↾s (lastS‘𝑣)))
120	119	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (ℂfld ↾s 𝑓) = (ℂfld ↾s (lastS‘𝑣)))
121	118, 120	breq12d 5115	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → ((ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓) ↔ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣))))
122	116, 121	bitrid 285	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (𝐸/FldExt𝐹 ↔ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣))))
123	114, 115	oveq12i 7410	. . . . . . . . . . . . . . . 16 ⊢ (𝐸[:]𝐹) = ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓))
124	118, 120	oveq12d 7416	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓)) = ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))))
125	123, 124	eqtrid 2811	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (𝐸[:]𝐹) = ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))))
126	125	eqeq1d 2766	. . . . . . . . . . . . . 14 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → ((𝐸[:]𝐹) = 2 ↔ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2))
127	122, 126	anbi12d 641	. . . . . . . . . . . . 13 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2) ↔ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2)))
128		constrextdg2.l	. . . . . . . . . . . . 13 ⊢ < = {⟨𝑓, 𝑒⟩ ∣ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 2)}
129	127, 128	brabga 5506	. . . . . . . . . . . 12 ⊢ (((lastS‘𝑣) ∈ V ∧ (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ∈ V) → ((lastS‘𝑣) < (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ↔ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2)))
130	129	biimpar 481	. . . . . . . . . . 11 ⊢ ((((lastS‘𝑣) ∈ V ∧ (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ∈ V) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))/FldExt(ℂfld ↾s (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2)) → (lastS‘𝑣) < (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
131	106, 107, 112, 113, 130	syl22anc 849	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (lastS‘𝑣) < (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
132	131	olcd 885	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝑣 = ∅ ∨ (lastS‘𝑣) < (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))))
133	104, 105, 132	chnccats1 18659	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) ∈ ( < Chain (SubDRing‘ℂfld)))
134	63	adantr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑣 ∈ Word (SubDRing‘ℂfld))
135	104	s1cld 14619	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩ ∈ Word (SubDRing‘ℂfld))
136		hashgt0 14403	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ( < Chain (SubDRing‘ℂfld)) ∧ 𝑣 ≠ ∅) → 0 < (♯‘𝑣))
137	41, 88, 136	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 0 < (♯‘𝑣))
138	137	adantr 484	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 0 < (♯‘𝑣))
139		ccatfv0 14599	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩ ∈ Word (SubDRing‘ℂfld) ∧ 0 < (♯‘𝑣)) → ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = (𝑣‘0))
140	134, 135, 138, 139	syl3anc 1392	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = (𝑣‘0))
141		simpllr 785	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝑣‘0) = ℚ)
142	140, 141	eqtrd 2799	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ)
143	45	adantr 484	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝐶‘𝑁) ⊆ (lastS‘𝑣))
144		ssun3 4134	. . . . . . . . . . . . 13 ⊢ ((𝐶‘𝑁) ⊆ (lastS‘𝑣) → (𝐶‘𝑁) ⊆ ((lastS‘𝑣) ∪ {𝑦}))
145	143, 144	syl 17	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝐶‘𝑁) ⊆ ((lastS‘𝑣) ∪ {𝑦}))
146		simplr 778	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))
147	146	unssbd 4148	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑔 ⊆ (lastS‘𝑣))
148		ssun3 4134	. . . . . . . . . . . . . 14 ⊢ (𝑔 ⊆ (lastS‘𝑣) → 𝑔 ⊆ ((lastS‘𝑣) ∪ {𝑦}))
149	147, 148	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → 𝑔 ⊆ ((lastS‘𝑣) ∪ {𝑦}))
150		ssun2 4133	. . . . . . . . . . . . . 14 ⊢ {𝑦} ⊆ ((lastS‘𝑣) ∪ {𝑦})
151	150	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → {𝑦} ⊆ ((lastS‘𝑣) ∪ {𝑦}))
152	149, 151	unssd 4146	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (𝑔 ∪ {𝑦}) ⊆ ((lastS‘𝑣) ∪ {𝑦}))
153	145, 152	unssd 4146	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ ((lastS‘𝑣) ∪ {𝑦}))
154	60, 62, 103	fldgenssid 33502	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((lastS‘𝑣) ∪ {𝑦}) ⊆ (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
155	153, 154	sstrd 3948	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
156		lswccats1 14650	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) ∈ (SubDRing‘ℂfld)) → (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)) = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
157	134, 104, 156	syl2anc 593	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)) = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
158	155, 157	sseqtrrd 3975	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))
159	142, 158	jca 519	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → (((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩))))
160	59, 133, 159	rspcedvdw 3586	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
161	73	ad5antr 744	. . . . . . . 8 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑁 ∈ On)
162	70, 108, 109, 90, 161, 45, 99	constrelextdg2 34046	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → (𝑦 ∈ (lastS‘𝑣) ∨ ((ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))[:](ℂfld ↾s (lastS‘𝑣))) = 2))
163	54, 160, 162	mpjaodan 971	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ (𝑣‘0) = ℚ) ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
164	163	anasss 470	. . . . 5 ⊢ (((((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
165	164	rexlimdva2 3167	. . . 4 ⊢ (((𝜑 ∧ 𝑔 ⊆ (𝐶‘suc 𝑁)) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔)) → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
166	165	anasss 470	. . 3 ⊢ ((𝜑 ∧ (𝑔 ⊆ (𝐶‘suc 𝑁) ∧ 𝑦 ∈ ((𝐶‘suc 𝑁) ∖ 𝑔))) → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ∃𝑢 ∈ ( < Chain (SubDRing‘ℂfld))((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
167		peano2 7872	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
168	71, 167	syl 17	. . . 4 ⊢ (𝜑 → suc 𝑁 ∈ ω)
169	70, 168	constrfin 34045	. . 3 ⊢ (𝜑 → (𝐶‘suc 𝑁) ∈ Fin)
170	4, 8, 19, 23, 35, 166, 169	findcard2d 9137	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)))
171		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)) → ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣))
172	171	unssbd 4148	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)) → (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣))
173	172	ex 416	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) → (((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣) → (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))
174	173	anim2d 621	. . 3 ⊢ ((𝜑 ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)) → ((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣))))
175	174	reximdva 3177	. 2 ⊢ (𝜑 → (∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)) → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣))))
176	170, 175	mpd 15	1 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ w3o 1098   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  ∃wrex 3088  {crab 3416  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4584  {cpr 4586   class class class wbr 5102  {copab 5164   ↦ cmpt 5183  Oncon0 6348  suc csuc 6350  ‘cfv 6523  (class class class)co 7398  ωcom 7848  reccrdg 8382  ℂcc 11073  ℝcr 11074  0cc0 11075  1c1 11076   + caddc 11078   · cmul 11080   < clt 11218   − cmin 11416  2c2 12274  ℚcq 12951  ♯chash 14345  Word cword 14528  lastSclsw 14577   ++ cconcat 14585  ⟨“cs1 14611  ∗ccj 15125  ℑcim 15127  abscabs 15263   ↾_s cress 17268   Chain cchn 18639  DivRingcdr 20781  Fieldcfield 20782  SubDRingcsdrg 20837  ℂ_fldccnfld 21426   fldGen cfldgen 33499  /_FldExtcfldext 33937  [:]cextdg 33939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-reg 9542  ax-inf2 9598  ax-ac2 10422  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153  ax-addf 11154  ax-mulf 11155

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-of 7662  df-ofr 7663  df-rpss 7708  df-om 7849  df-1st 7972  df-2nd 7973  df-supp 8143  df-tpos 8208  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-oadd 8443  df-er 8680  df-ec 8682  df-qs 8686  df-map 8812  df-pm 8813  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-fin 8933  df-fsupp 9310  df-sup 9390  df-inf 9391  df-oi 9460  df-r1 9724  df-rank 9725  df-dju 9861  df-card 9899  df-acn 9902  df-ac 10074  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-div 11847  df-nn 12213  df-2 12282  df-3 12283  df-4 12284  df-5 12285  df-6 12286  df-7 12287  df-8 12288  df-9 12289  df-n0 12484  df-xnn0 12557  df-z 12571  df-dec 12691  df-uz 12842  df-rp 12996  df-ico 13357  df-fz 13515  df-fzo 13662  df-seq 14017  df-exp 14077  df-hash 14346  df-word 14529  df-lsw 14578  df-concat 14586  df-s1 14612  df-cj 15128  df-re 15129  df-im 15130  df-sqrt 15264  df-abs 15265  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17248  df-ress 17269  df-plusg 17301  df-mulr 17302  df-starv 17303  df-sca 17304  df-vsca 17305  df-ip 17306  df-tset 17307  df-ple 17308  df-ocomp 17309  df-ds 17310  df-unif 17311  df-hom 17312  df-cco 17313  df-0g 17472  df-gsum 17473  df-prds 17478  df-pws 17480  df-imas 17540  df-qus 17541  df-mre 17616  df-mrc 17617  df-mri 17618  df-acs 17619  df-proset 18328  df-drs 18329  df-poset 18347  df-ipo 18562  df-chn 18640  df-mgm 18676  df-sgrp 18755  df-mnd 18771  df-mhm 18819  df-submnd 18820  df-grp 18980  df-minusg 18981  df-sbg 18982  df-mulg 19112  df-subg 19167  df-nsg 19168  df-eqg 19169  df-ghm 19256  df-gim 19301  df-cntz 19359  df-oppg 19388  df-lsm 19678  df-cmn 19824  df-abl 19825  df-mgp 20189  df-rng 20201  df-ur 20234  df-srg 20239  df-ring 20287  df-cring 20288  df-oppr 20388  df-dvdsr 20408  df-unit 20409  df-irred 20410  df-invr 20439  df-dvr 20452  df-rhm 20523  df-nzr 20565  df-subrng 20598  df-subrg 20622  df-rlreg 20746  df-domn 20747  df-idom 20748  df-drng 20783  df-field 20784  df-sdrg 20838  df-lmod 20931  df-lss 21001  df-lsp 21041  df-lmhm 21091  df-lmim 21092  df-lmic 21093  df-lbs 21144  df-lvec 21172  df-sra 21242  df-rgmod 21243  df-lidl 21280  df-rsp 21281  df-2idl 21322  df-lpidl 21394  df-lpir 21395  df-pid 21409  df-cnfld 21427  df-dsmm 21786  df-frlm 21801  df-uvc 21837  df-lindf 21860  df-linds 21861  df-assa 21907  df-asp 21908  df-ascl 21909  df-psr 21963  df-mvr 21964  df-mpl 21965  df-opsr 21967  df-evls 22129  df-evl 22130  df-psr1 22244  df-vr1 22245  df-ply1 22246  df-coe1 22247  df-evls1 22380  df-evl1 22381  df-mdeg 26117  df-deg1 26118  df-mon1 26193  df-uc1p 26194  df-q1p 26195  df-r1p 26196  df-ig1p 26197  df-fldgen 33500  df-mxidl 33650  df-dim 33899  df-fldext 33940  df-extdg 33941  df-irng 33983  df-minply 33999

This theorem is referenced by:  constrextdg2  34048