Theorem constrextdg2lem 33892

Description: Lemma for constrextdg2 33893. (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 4102	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ ∅))
2	1	sseq1d 3953	. . . . 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 3161	. . 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 4102	. . . . . 6 ⊢ (𝑖 = 𝑔 → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ 𝑔))
6	5	sseq1d 3953	. . . . 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 3161	. . 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 6839	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
10	9	eqeq1d 2738	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
11		fveq2 6840	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
12	11	sseq2d 3954	. . . . . 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 3216	. . . 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 4102	. . . . . . 7 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})))
16	15	sseq1d 3953	. . . . . 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 3161	. . . 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 4102	. . . . . 6 ⊢ (𝑖 = (𝐶‘suc 𝑁) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)))
21	20	sseq1d 3953	. . . . 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 3161	. . 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 6839	. . . . . 6 ⊢ (𝑣 = 𝑅 → (𝑣‘0) = (𝑅‘0))
25	24	eqeq1d 2738	. . . . 5 ⊢ (𝑣 = 𝑅 → ((𝑣‘0) = ℚ ↔ (𝑅‘0) = ℚ))
26		fveq2 6840	. . . . . 6 ⊢ (𝑣 = 𝑅 → (lastS‘𝑣) = (lastS‘𝑅))
27	26	sseq2d 3954	. . . . 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 4334	. . . . . 6 ⊢ ((𝐶‘𝑁) ∪ ∅) = (𝐶‘𝑁)
32		constrextdg2lem.3	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))
33	31, 32	eqsstrid 3960	. . . . 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 3567	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
36		fveq1 6839	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢‘0) = (𝑣‘0))
37	36	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢‘0) = ℚ ↔ (𝑣‘0) = ℚ))
38		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (lastS‘𝑢) = (lastS‘𝑣))
39	38	sseq2d 3954	. . . . . . . . 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 4133	. . . . . . . . . . 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 4134	. . . . . . . . . . 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 4730	. . . . . . . . . . 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 4132	. . . . . . . . . 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 4132	. . . . . . . . 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 3567	. . . . . . 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 6839	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (𝑢‘0) = ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0))
56	55	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → ((𝑢‘0) = ℚ ↔ ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ))
57		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (lastS‘𝑢) = (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))
58	57	sseq2d 3954	. . . . . . . . 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 21356	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
61		cndrng 21381	. . . . . . . . . . 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 18574	. . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = (lastS‘∅))
66		lsw0g 14528	. . . . . . . . . . . . . . . . . . 19 ⊢ (lastS‘∅) = ∅
67	65, 66	eqtrdi 2787	. . . . . . . . . . . . . . . . . 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 4118	. . . . . . . . . . . . . . . . . . . . . 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 7823	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ On)
74	70, 73	constr01 33886	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
75		c0ex 11138	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
76	75	prnz 4721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {0, 1} ≠ ∅
77		ssn0 4344	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({0, 1} ⊆ (𝐶‘𝑁) ∧ {0, 1} ≠ ∅) → (𝐶‘𝑁) ≠ ∅)
78	74, 76, 77	sylancl 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐶‘𝑁) ≠ ∅)
79		ssn0 4344	. . . . . . . . . . . . . . . . . . . . . 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 4344	. . . . . . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . . . . . 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 2939	. . . . . . . . . . . . . . . 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 32522	. . . . . . . . . . . . . 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 14530	. . . . . . . . . . . . . 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 20770	. . . . . . . . . . . 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 7764	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
95	73, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → suc 𝑁 ∈ On)
96	70, 95	constrsscn 33884	. . . . . . . . . . . . . 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 3901	. . . . . . . . . . . . . 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 3922	. . . . . . . . . . . 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 4730	. . . . . . . . . . 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 4132	. . . . . . . . . 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 33375	. . . . . . . . 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 3453	. . . . . . . . . . 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 3453	. . . . . . . . . . 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 2736	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
109		eqid 2736	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
110		cnfldfld 33402	. . . . . . . . . . . . 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 33812	. . . . . . . . . . 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 5094	. . . . . . . . . . . . . . 15 ⊢ (𝐸/FldExt𝐹 ↔ (ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓))
117		oveq2 7375	. . . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . . 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 7379	. . . . . . . . . . . . . . . 16 ⊢ (𝐸[:]𝐹) = ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓))
124	118, 120	oveq12d 7385	. . . . . . . . . . . . . . . 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 2783	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 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 5489	. . . . . . . . . . . 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 18591	. . . . . . . 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 14566	. . . . . . . . . . 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 14350	. . . . . . . . . . . . 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 14546	. . . . . . . . . . 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 2771	. . . . . . . . 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 4120	. . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . 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 4120	. . . . . . . . . . . . . 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 4119	. . . . . . . . . . . . . 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 4132	. . . . . . . . . . . 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 4132	. . . . . . . . . . 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 33374	. . . . . . . . . . 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 3932	. . . . . . . . . 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 14597	. . . . . . . . . . 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 3959	. . . . . . . . 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 3567	. . . . . . 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 33891	. . . . . . 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 3140	. . . 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 7841	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
168	71, 167	syl 17	. . . 4 ⊢ (𝜑 → suc 𝑁 ∈ ω)
169	70, 168	constrfin 33890	. . 3 ⊢ (𝜑 → (𝐶‘suc 𝑁) ∈ Fin)
170	4, 8, 19, 23, 35, 166, 169	findcard2d 9101	. 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 4134	. . . . 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 3150	. 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 2932  ∃wrex 3061  {crab 3389  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  {csn 4567  {cpr 4569   class class class wbr 5085  {copab 5147   ↦ cmpt 5166  Oncon0 6323  suc csuc 6325  ‘cfv 6498  (class class class)co 7367  ωcom 7817  reccrdg 8348  ℂcc 11036  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11179   − cmin 11377  2c2 12236  ℚcq 12898  ♯chash 14292  Word cword 14475  lastSclsw 14524   ++ cconcat 14532  ⟨“cs1 14558  ∗ccj 15058  ℑcim 15060  abscabs 15196   ↾_s cress 17200   Chain cchn 18571  DivRingcdr 20706  Fieldcfield 20707  SubDRingcsdrg 20763  ℂ_fldccnfld 21352   fldGen cfldgen 33371  /_FldExtcfldext 33782  [:]cextdg 33784

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117  ax-mulf 11118

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-rpss 7677  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-ec 8645  df-qs 8649  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-inf 9356  df-oi 9425  df-r1 9688  df-rank 9689  df-dju 9825  df-card 9863  df-acn 9866  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-ico 13304  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ocomp 17241  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-imas 17472  df-qus 17473  df-mre 17548  df-mrc 17549  df-mri 17550  df-acs 17551  df-proset 18260  df-drs 18261  df-poset 18279  df-ipo 18494  df-chn 18572  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-nsg 19100  df-eqg 19101  df-ghm 19188  df-gim 19234  df-cntz 19292  df-oppg 19321  df-lsm 19611  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-irred 20339  df-invr 20368  df-dvr 20381  df-rhm 20452  df-nzr 20490  df-subrng 20523  df-subrg 20547  df-rlreg 20671  df-domn 20672  df-idom 20673  df-drng 20708  df-field 20709  df-sdrg 20764  df-lmod 20857  df-lss 20927  df-lsp 20967  df-lmhm 21017  df-lmim 21018  df-lmic 21019  df-lbs 21070  df-lvec 21098  df-sra 21168  df-rgmod 21169  df-lidl 21206  df-rsp 21207  df-2idl 21248  df-lpidl 21320  df-lpir 21321  df-pid 21335  df-cnfld 21353  df-dsmm 21712  df-frlm 21727  df-uvc 21763  df-lindf 21786  df-linds 21787  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-opsr 21893  df-evls 22052  df-evl 22053  df-psr1 22143  df-vr1 22144  df-ply1 22145  df-coe1 22146  df-evls1 22280  df-evl1 22281  df-mdeg 26020  df-deg1 26021  df-mon1 26096  df-uc1p 26097  df-q1p 26098  df-r1p 26099  df-ig1p 26100  df-fldgen 33372  df-mxidl 33520  df-dim 33744  df-fldext 33785  df-extdg 33786  df-irng 33828  df-minply 33844

This theorem is referenced by:  constrextdg2  33893