Theorem constrextdg2lem 33731

Description: Lemma for constrextdg2 33732 (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 4121	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ ∅))
2	1	sseq1d 3975	. . . . 5 ⊢ (𝑖 = ∅ → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
3	2	anbi2d 630	. . . 4 ⊢ (𝑖 = ∅ → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣))))
4	3	rexbidv 3157	. . 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 4121	. . . . . 6 ⊢ (𝑖 = 𝑔 → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ 𝑔))
6	5	sseq1d 3975	. . . . 5 ⊢ (𝑖 = 𝑔 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)))
7	6	anbi2d 630	. . . 4 ⊢ (𝑖 = 𝑔 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))))
8	7	rexbidv 3157	. . 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 2731	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
11		fveq2 6840	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
12	11	sseq2d 3976	. . . . . 6 ⊢ (𝑣 = 𝑢 → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)))
13	10, 12	anbi12d 632	. . . . 5 ⊢ (𝑣 = 𝑢 → (((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢))))
14	13	cbvrexvw 3214	. . . 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 4121	. . . . . . 7 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})))
16	15	sseq1d 3975	. . . . . 6 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)))
17	16	anbi2d 630	. . . . 5 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑢)) ↔ ((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢))))
18	17	rexbidv 3157	. . . 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 4121	. . . . . 6 ⊢ (𝑖 = (𝐶‘suc 𝑁) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)))
21	20	sseq1d 3975	. . . . 5 ⊢ (𝑖 = (𝐶‘suc 𝑁) → (((𝐶‘𝑁) ∪ 𝑖) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)) ⊆ (lastS‘𝑣)))
22	21	anbi2d 630	. . . 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 3157	. . 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 2731	. . . . 5 ⊢ (𝑣 = 𝑅 → ((𝑣‘0) = ℚ ↔ (𝑅‘0) = ℚ))
26		fveq2 6840	. . . . . 6 ⊢ (𝑣 = 𝑅 → (lastS‘𝑣) = (lastS‘𝑅))
27	26	sseq2d 3976	. . . . 5 ⊢ (𝑣 = 𝑅 → (((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣) ↔ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑅)))
28	25, 27	anbi12d 632	. . . 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 4353	. . . . . 6 ⊢ ((𝐶‘𝑁) ∪ ∅) = (𝐶‘𝑁)
32		constrextdg2lem.3	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))
33	31, 32	eqsstrid 3982	. . . . 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 3588	. . 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 2731	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢‘0) = ℚ ↔ (𝑣‘0) = ℚ))
38		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (lastS‘𝑢) = (lastS‘𝑣))
39	38	sseq2d 3976	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢) ↔ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣)))
40	37, 39	anbi12d 632	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (((𝑢‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑢)) ↔ ((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})) ⊆ (lastS‘𝑣))))
41		simpllr 775	. . . . . . . . 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 775	. . . . . . . . 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 4152	. . . . . . . . . . 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 768	. . . . . . . . . . . 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 4153	. . . . . . . . . . 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 4769	. . . . . . . . . . 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 4151	. . . . . . . . . 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 4151	. . . . . . . . 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 3588	. . . . . . 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 2731	. . . . . . . . 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 3976	. . . . . . . . 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 632	. . . . . . . 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 21300	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
61		cndrng 21340	. . . . . . . . . . 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 32979	. . . . . . . . . . . . . 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 14507	. . . . . . . . . . . . . . . . . . 19 ⊢ (lastS‘∅) = ∅
67	65, 66	eqtrdi 2780	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = ∅)
68		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣))
69		ssun1 4137	. . . . . . . . . . . . . . . . . . . . . 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 7828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ On)
74	70, 73	constr01 33725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
75		c0ex 11144	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
76	75	prnz 4737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {0, 1} ≠ ∅
77		ssn0 4363	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({0, 1} ⊆ (𝐶‘𝑁) ∧ {0, 1} ≠ ∅) → (𝐶‘𝑁) ≠ ∅)
78	74, 76, 77	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐶‘𝑁) ≠ ∅)
79		ssn0 4363	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐶‘𝑁) ⊆ ((𝐶‘𝑁) ∪ 𝑔) ∧ (𝐶‘𝑁) ≠ ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
80	69, 78, 79	sylancr 587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
81	80	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅)
82		ssn0 4363	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣) ∧ ((𝐶‘𝑁) ∪ 𝑔) ≠ ∅) → (lastS‘𝑣) ≠ ∅)
83	68, 81, 82	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) ≠ ∅)
84	83	neneqd 2930	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → ¬ (lastS‘𝑣) = ∅)
85	67, 84	pm2.65da 816	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → ¬ 𝑣 = ∅)
86	85	neqned 2932	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) → 𝑣 ≠ ∅)
87	86	ad4antr 732	. . . . . . . . . . . . . . 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 32431	. . . . . . . . . . . . . 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 14509	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word (SubDRing‘ℂfld) ∧ 𝑣 ≠ ∅) → (lastS‘𝑣) ∈ (SubDRing‘ℂfld))
90	63, 88, 89	syl2anc 584	. . . . . . . . . . . . 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 20713	. . . . . . . . . . . 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 33723	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐶‘suc 𝑁) ⊆ ℂ)
97	96	ad6antr 736	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . 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 3944	. . . . . . . . . . . 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 4769	. . . . . . . . . . 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 4151	. . . . . . . . . 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 33280	. . . . . . . . 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 3468	. . . . . . . . . . 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 3468	. . . . . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
109		eqid 2729	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
110		cnfldfld 33307	. . . . . . . . . . . . 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 33656	. . . . . . . . . . 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 5111	. . . . . . . . . . . . . . 15 ⊢ (𝐸/FldExt𝐹 ↔ (ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓))
117		oveq2 7377	. . . . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . 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 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 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 7381	. . . . . . . . . . . . . . . 16 ⊢ (𝐸[:]𝐹) = ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓))
124	118, 120	oveq12d 7387	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 838	. . . . . . . . . 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 874	. . . . . . . . 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 32987	. . . . . . . 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 14544	. . . . . . . . . . 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 14329	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ( < Chain(SubDRing‘ℂfld)) ∧ 𝑣 ≠ ∅) → 0 < (♯‘𝑣))
137	41, 88, 136	syl2anc 584	. . . . . . . . . . . 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 14524	. . . . . . . . . . 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 1373	. . . . . . . . . 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 775	. . . . . . . . . 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 2764	. . . . . . . . 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 4139	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 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 4153	. . . . . . . . . . . . . 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 4139	. . . . . . . . . . . . . 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 4138	. . . . . . . . . . . . . 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 4151	. . . . . . . . . . . 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 4151	. . . . . . . . . . 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 33279	. . . . . . . . . . 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 3954	. . . . . . . . . 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 14575	. . . . . . . . . . 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 584	. . . . . . . . . 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 3981	. . . . . . . . 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 3588	. . . . . . 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 734	. . . . . . . 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 33730	. . . . . . 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 960	. . . . . 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 3136	. . . 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 7846	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
168	71, 167	syl 17	. . . 4 ⊢ (𝜑 → suc 𝑁 ∈ ω)
169	70, 168	constrfin 33729	. . 3 ⊢ (𝜑 → (𝐶‘suc 𝑁) ∈ Fin)
170	4, 8, 19, 23, 35, 166, 169	findcard2d 9107	. 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 4153	. . . . 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 612	. . 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 3146	. 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 1085   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053  {crab 3402  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4585  {cpr 4587   class class class wbr 5102  {copab 5164   ↦ cmpt 5183  Oncon0 6320  suc csuc 6322  ‘cfv 6499  (class class class)co 7369  ωcom 7822  reccrdg 8354  ℂcc 11042  ℝcr 11043  0cc0 11044  1c1 11045   + caddc 11047   · cmul 11049   < clt 11184   − cmin 11381  2c2 12217  ℚcq 12883  ♯chash 14271  Word cword 14454  lastSclsw 14503   ++ cconcat 14511  ⟨“cs1 14536  ∗ccj 15038  ℑcim 15040  abscabs 15176   ↾_s cress 17176  DivRingcdr 20649  Fieldcfield 20650  SubDRingcsdrg 20706  ℂ_fldccnfld 21296  Chaincchn 32976   fldGen cfldgen 33276  /_FldExtcfldext 33627  [:]cextdg 33629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-reg 9521  ax-inf2 9570  ax-ac2 10392  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122  ax-addf 11123  ax-mulf 11124

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  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 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-ofr 7634  df-rpss 7679  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-er 8648  df-ec 8650  df-qs 8654  df-map 8778  df-pm 8779  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-sup 9369  df-inf 9370  df-oi 9439  df-r1 9693  df-rank 9694  df-dju 9830  df-card 9868  df-acn 9871  df-ac 10045  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-xnn0 12492  df-z 12506  df-dec 12626  df-uz 12770  df-rp 12928  df-ico 13288  df-fz 13445  df-fzo 13592  df-seq 13943  df-exp 14003  df-hash 14272  df-word 14455  df-lsw 14504  df-concat 14512  df-s1 14537  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ocomp 17217  df-ds 17218  df-unif 17219  df-hom 17220  df-cco 17221  df-0g 17380  df-gsum 17381  df-prds 17386  df-pws 17388  df-imas 17447  df-qus 17448  df-mre 17523  df-mrc 17524  df-mri 17525  df-acs 17526  df-proset 18235  df-drs 18236  df-poset 18254  df-ipo 18469  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-mhm 18692  df-submnd 18693  df-grp 18850  df-minusg 18851  df-sbg 18852  df-mulg 18982  df-subg 19037  df-nsg 19038  df-eqg 19039  df-ghm 19127  df-gim 19173  df-cntz 19231  df-oppg 19260  df-lsm 19550  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-srg 20107  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-irred 20279  df-invr 20308  df-dvr 20321  df-rhm 20392  df-nzr 20433  df-subrng 20466  df-subrg 20490  df-rlreg 20614  df-domn 20615  df-idom 20616  df-drng 20651  df-field 20652  df-sdrg 20707  df-lmod 20800  df-lss 20870  df-lsp 20910  df-lmhm 20961  df-lmim 20962  df-lmic 20963  df-lbs 21014  df-lvec 21042  df-sra 21112  df-rgmod 21113  df-lidl 21150  df-rsp 21151  df-2idl 21192  df-lpidl 21264  df-lpir 21265  df-pid 21279  df-cnfld 21297  df-dsmm 21674  df-frlm 21689  df-uvc 21725  df-lindf 21748  df-linds 21749  df-assa 21795  df-asp 21796  df-ascl 21797  df-psr 21851  df-mvr 21852  df-mpl 21853  df-opsr 21855  df-evls 22014  df-evl 22015  df-psr1 22097  df-vr1 22098  df-ply1 22099  df-coe1 22100  df-evls1 22235  df-evl1 22236  df-mdeg 25993  df-deg1 25994  df-mon1 26069  df-uc1p 26070  df-q1p 26071  df-r1p 26072  df-ig1p 26073  df-chn 32977  df-fldgen 33277  df-mxidl 33424  df-dim 33588  df-fldext 33630  df-extdg 33631  df-irng 33672  df-minply 33683

This theorem is referenced by:  constrextdg2  33732