Theorem constrextdg2lem 33932

Description: Lemma for constrextdg2 33933. (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 4116	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ ∅))
2	1	sseq1d 3967	. . . . 5 ⊢ (𝑖 = ∅ → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
3	2	anbi2d 631	. . . 4 ⊢ (𝑖 = ∅ → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣))))
4	3	rexbidv 3162	. . 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 4116	. . . . . 6 ⊢ (𝑖 = 𝑔 → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ 𝑔))
6	5	sseq1d 3967	. . . . 5 ⊢ (𝑖 = 𝑔 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)))
7	6	anbi2d 631	. . . 4 ⊢ (𝑖 = 𝑔 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))))
8	7	rexbidv 3162	. . 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 6843	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
10	9	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
11		fveq2 6844	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
12	11	sseq2d 3968	. . . . . 6 ⊢ (𝑣 = 𝑢 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)))
13	10, 12	anbi12d 633	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢))))
14	13	cbvrexvw 3217	. . . 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 4116	. . . . . . 7 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})))
16	15	sseq1d 3967	. . . . . 6 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
17	16	anbi2d 631	. . . . 5 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
18	17	rexbidv 3162	. . . 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 283	. . 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 4116	. . . . . 6 ⊢ (𝑖 = (𝐶‘suc 𝑁) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)))
21	20	sseq1d 3967	. . . . 5 ⊢ (𝑖 = (𝐶‘suc 𝑁) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)))
22	21	anbi2d 631	. . . 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 3162	. . 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 6843	. . . . . 6 ⊢ (𝑣 = 𝑅 → (𝑣‘0) = (𝑅‘0))
25	24	eqeq1d 2739	. . . . 5 ⊢ (𝑣 = 𝑅 → ((𝑣‘0) = ℚ ↔ (𝑅‘0) = ℚ))
26		fveq2 6844	. . . . . 6 ⊢ (𝑣 = 𝑅 → (lastS‘𝑣) = (lastS‘𝑅))
27	26	sseq2d 3968	. . . . 5 ⊢ (𝑣 = 𝑅 → (((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅)))
28	25, 27	anbi12d 633	. . . 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 4348	. . . . . 6 ⊢ ((𝐶‘𝑁) ∪ ∅) = (𝐶‘𝑁)
32		constrextdg2lem.3	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))
33	31, 32	eqsstrid 3974	. . . . 5 ⊢ (𝜑 → ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅))
34	30, 33	jca 511	. . . 4 ⊢ (𝜑 → ((𝑅‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅)))
35	28, 29, 34	rspcedvdw 3581	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
36		fveq1 6843	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢‘0) = (𝑣‘0))
37	36	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢‘0) = ℚ ↔ (𝑣‘0) = ℚ))
38		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (lastS‘𝑢) = (lastS‘𝑣))
39	38	sseq2d 3968	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣)))
40	37, 39	anbi12d 633	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣))))
41		simpllr 776	. . . . . . . . 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 480	. . . . . . . 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 776	. . . . . . . . 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 484	. . . . . . . . . . . 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 480	. . . . . . . . . 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 769	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 4767	. . . . . . . . . . 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 511	. . . . . . . 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 3581	. . . . . . 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 6843	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (𝑢‘0) = ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0))
56	55	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → ((𝑢‘0) = ℚ ↔ ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ))
57		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (lastS‘𝑢) = (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))
58	57	sseq2d 3968	. . . . . . . . 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 633	. . . . . . . 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 21330	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
61		cndrng 21370	. . . . . . . . . . 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 18545	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → 𝑣 = ∅)
65	64	fveq2d 6848	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = (lastS‘∅))
66		lsw0g 14503	. . . . . . . . . . . . . . . . . . 19 ⊢ (lastS‘∅) = ∅
67	65, 66	eqtrdi 2788	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = ∅)
68		simplr 769	. . . . . . . . . . . . . . . . . . . 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 7826	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ On)
74	70, 73	constr01 33926	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
75		c0ex 11140	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
76	75	prnz 4736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {0, 1} ≠ ∅
77		ssn0 4358	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({0, 1} ⊆ (𝐶‘𝑁) ∧ {0, 1} ≠ ∅) → (𝐶‘𝑁) ≠ ∅)
78	74, 76, 77	sylancl 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐶‘𝑁) ≠ ∅)
79		ssn0 4358	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶‘𝑁) ⊆ ((𝐶‘𝑁) ∪ 𝑔) ∧ (𝐶‘𝑁) ≠ ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
80	69, 78, 79	sylancr 588	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
81	80	ad2antrr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
82		ssn0 4358	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣) ∧ ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅) → (lastS‘𝑣) ≠ ∅)
83	68, 81, 82	syl2anc 585	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) ≠ ∅)
84	83	neneqd 2938	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ¬ (lastS‘𝑣) = ∅)
85	67, 84	pm2.65da 817	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ¬ 𝑣 = ∅)
86	85	neqned 2940	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ≠ ∅)
87	86	ad4antr 733	. . . . . . . . . . . . . . 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 32545	. . . . . . . . . . . . . 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 14505	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
90	63, 88, 89	syl2anc 585	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 20743	. . . . . . . . . . . 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 7767	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
95	73, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → suc 𝑁 ∈ On)
96	70, 95	constrsscn 33924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘suc 𝑁) ⊆ ℂ)
97	96	ad6antr 737	. . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 3936	. . . . . . . . . . . 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 4767	. . . . . . . . . . 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 33414	. . . . . . . . 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 480	. . . . . . . . 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 3466	. . . . . . . . . . 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 3466	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
109		eqid 2737	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
110		cnfldfld 33441	. . . . . . . . . . . . 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 33852	. . . . . . . . . . 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 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) → ((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 5109	. . . . . . . . . . . . . . 15 ⊢ (𝐸/FldExt𝐹 ↔ (ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓))
117		oveq2 7378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})) → (ℂfld ↾s 𝑒) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))))
118	117	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (ℂfld ↾s 𝑒) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))))
119		oveq2 7378	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (lastS‘𝑣) → (ℂfld ↾s 𝑓) = (ℂfld ↾s (lastS‘𝑣)))
120	119	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 = (lastS‘𝑣) ∧ 𝑒 = (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) → (ℂfld ↾s 𝑓) = (ℂfld ↾s (lastS‘𝑣)))
121	118, 120	breq12d 5113	. . . . . . . . . . . . . . 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 283	. . . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . . . 16 ⊢ (𝐸[:]𝐹) = ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓))
124	118, 120	oveq12d 7388	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . 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 633	. . . . . . . . . . . . 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 5492	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 839	. . . . . . . . . 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 875	. . . . . . . . 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 18562	. . . . . . . 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 480	. . . . . . . . . . 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 14541	. . . . . . . . . . 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 14325	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ( < Chain (SubDRing‘ℂfld)) ∧ 𝑣 ≠ ∅) → 0 < (♯‘𝑣))
137	41, 88, 136	syl2anc 585	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 14521	. . . . . . . . . . 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 1374	. . . . . . . . . 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 776	. . . . . . . . . 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 2772	. . . . . . . . 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 480	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . 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 33413	. . . . . . . . . . 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 3946	. . . . . . . . . 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 14572	. . . . . . . . . . 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 585	. . . . . . . . . 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 3973	. . . . . . . . 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 511	. . . . . . . 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 3581	. . . . . . 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 735	. . . . . . . 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 33931	. . . . . . 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 961	. . . . . 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 466	. . . . 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 3141	. . . 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 466	. . 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 7844	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
168	71, 167	syl 17	. . . 4 ⊢ (𝜑 → suc 𝑁 ∈ ω)
169	70, 168	constrfin 33930	. . 3 ⊢ (𝜑 → (𝐶‘suc 𝑁) ∈ Fin)
170	4, 8, 19, 23, 35, 166, 169	findcard2d 9105	. 2 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)))
171		simpr 484	. . . . . 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 412	. . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ ( < Chain (SubDRing‘ℂfld))) → (((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣) → (𝐶‘suc 𝑁) ⊆ (lastS‘𝑣)))
174	173	anim2d 613	. . 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 3151	. 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 395   ∨ w3o 1086   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  {csn 4582  {cpr 4584   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  Oncon0 6327  suc csuc 6329  ‘cfv 6502  (class class class)co 7370  ωcom 7820  reccrdg 8352  ℂcc 11038  ℝcr 11039  0cc0 11040  1c1 11041   + caddc 11043   · cmul 11045   < clt 11180   − cmin 11378  2c2 12214  ℚcq 12875  ♯chash 14267  Word cword 14450  lastSclsw 14499   ++ cconcat 14507  ⟨“cs1 14533  ∗ccj 15033  ℑcim 15035  abscabs 15171   ↾_s cress 17171   Chain cchn 18542  DivRingcdr 20679  Fieldcfield 20680  SubDRingcsdrg 20736  ℂ_fldccnfld 21326   fldGen cfldgen 33410  /_FldExtcfldext 33822  [:]cextdg 33824

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-reg 9511  ax-inf2 9564  ax-ac2 10387  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118  ax-addf 11119  ax-mulf 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-of 7634  df-ofr 7635  df-rpss 7680  df-om 7821  df-1st 7945  df-2nd 7946  df-supp 8115  df-tpos 8180  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-er 8647  df-ec 8649  df-qs 8653  df-map 8779  df-pm 8780  df-ixp 8850  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-fsupp 9279  df-sup 9359  df-inf 9360  df-oi 9429  df-r1 9690  df-rank 9691  df-dju 9827  df-card 9865  df-acn 9868  df-ac 10040  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-4 12224  df-5 12225  df-6 12226  df-7 12227  df-8 12228  df-9 12229  df-n0 12416  df-xnn0 12489  df-z 12503  df-dec 12622  df-uz 12766  df-rp 12920  df-ico 13281  df-fz 13438  df-fzo 13585  df-seq 13939  df-exp 13999  df-hash 14268  df-word 14451  df-lsw 14500  df-concat 14508  df-s1 14534  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-struct 17088  df-sets 17105  df-slot 17123  df-ndx 17135  df-base 17151  df-ress 17172  df-plusg 17204  df-mulr 17205  df-starv 17206  df-sca 17207  df-vsca 17208  df-ip 17209  df-tset 17210  df-ple 17211  df-ocomp 17212  df-ds 17213  df-unif 17214  df-hom 17215  df-cco 17216  df-0g 17375  df-gsum 17376  df-prds 17381  df-pws 17383  df-imas 17443  df-qus 17444  df-mre 17519  df-mrc 17520  df-mri 17521  df-acs 17522  df-proset 18231  df-drs 18232  df-poset 18250  df-ipo 18465  df-chn 18543  df-mgm 18579  df-sgrp 18658  df-mnd 18674  df-mhm 18722  df-submnd 18723  df-grp 18883  df-minusg 18884  df-sbg 18885  df-mulg 19015  df-subg 19070  df-nsg 19071  df-eqg 19072  df-ghm 19159  df-gim 19205  df-cntz 19263  df-oppg 19292  df-lsm 19582  df-cmn 19728  df-abl 19729  df-mgp 20093  df-rng 20105  df-ur 20134  df-srg 20139  df-ring 20187  df-cring 20188  df-oppr 20290  df-dvdsr 20310  df-unit 20311  df-irred 20312  df-invr 20341  df-dvr 20354  df-rhm 20425  df-nzr 20463  df-subrng 20496  df-subrg 20520  df-rlreg 20644  df-domn 20645  df-idom 20646  df-drng 20681  df-field 20682  df-sdrg 20737  df-lmod 20830  df-lss 20900  df-lsp 20940  df-lmhm 20991  df-lmim 20992  df-lmic 20993  df-lbs 21044  df-lvec 21072  df-sra 21142  df-rgmod 21143  df-lidl 21180  df-rsp 21181  df-2idl 21222  df-lpidl 21294  df-lpir 21295  df-pid 21309  df-cnfld 21327  df-dsmm 21704  df-frlm 21719  df-uvc 21755  df-lindf 21778  df-linds 21779  df-assa 21825  df-asp 21826  df-ascl 21827  df-psr 21882  df-mvr 21883  df-mpl 21884  df-opsr 21886  df-evls 22046  df-evl 22047  df-psr1 22137  df-vr1 22138  df-ply1 22139  df-coe1 22140  df-evls1 22276  df-evl1 22277  df-mdeg 26033  df-deg1 26034  df-mon1 26109  df-uc1p 26110  df-q1p 26111  df-r1p 26112  df-ig1p 26113  df-fldgen 33411  df-mxidl 33559  df-dim 33783  df-fldext 33825  df-extdg 33826  df-irng 33868  df-minply 33884

This theorem is referenced by:  constrextdg2  33933