Theorem constrextdg2lem 33854

Description: Lemma for constrextdg2 33855 (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 4112	. . . . . 6 ⊢ (𝑖 = ∅ → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ ∅))
2	1	sseq1d 3963	. . . . 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 3158	. . 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 4112	. . . . . 6 ⊢ (𝑖 = 𝑔 → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ 𝑔))
6	5	sseq1d 3963	. . . . 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 3158	. . 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 6831	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (𝑣‘0) = (𝑢‘0))
10	9	eqeq1d 2736	. . . . . 6 ⊢ (𝑣 = 𝑢 → ((𝑣‘0) = ℚ ↔ (𝑢‘0) = ℚ))
11		fveq2 6832	. . . . . . 7 ⊢ (𝑣 = 𝑢 → (lastS‘𝑣) = (lastS‘𝑢))
12	11	sseq2d 3964	. . . . . 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 3213	. . . 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 4112	. . . . . . 7 ⊢ (𝑖 = (𝑔 ∪ {𝑦}) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝑔 ∪ {𝑦})))
16	15	sseq1d 3963	. . . . . 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 3158	. . . 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 4112	. . . . . 6 ⊢ (𝑖 = (𝐶‘suc 𝑁) → ((𝐶‘𝑁) ∪ 𝑖) = ((𝐶‘𝑁) ∪ (𝐶‘suc 𝑁)))
21	20	sseq1d 3963	. . . . 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 3158	. . 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 6831	. . . . . 6 ⊢ (𝑣 = 𝑅 → (𝑣‘0) = (𝑅‘0))
25	24	eqeq1d 2736	. . . . 5 ⊢ (𝑣 = 𝑅 → ((𝑣‘0) = ℚ ↔ (𝑅‘0) = ℚ))
26		fveq2 6832	. . . . . 6 ⊢ (𝑣 = 𝑅 → (lastS‘𝑣) = (lastS‘𝑅))
27	26	sseq2d 3964	. . . . 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 4344	. . . . . 6 ⊢ ((𝐶‘𝑁) ∪ ∅) = (𝐶‘𝑁)
32		constrextdg2lem.3	. . . . . 6 ⊢ (𝜑 → (𝐶‘𝑁) ⊆ (lastS‘𝑅))
33	31, 32	eqsstrid 3970	. . . . 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 3577	. . 3 ⊢ (𝜑 → ∃𝑣 ∈ ( < Chain (SubDRing‘ℂfld))((𝑣‘0) = ℚ ∧ ((𝐶‘𝑁) ∪ ∅) ⊆ (lastS‘𝑣)))
36		fveq1 6831	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢‘0) = (𝑣‘0))
37	36	eqeq1d 2736	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢‘0) = ℚ ↔ (𝑣‘0) = ℚ))
38		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (lastS‘𝑢) = (lastS‘𝑣))
39	38	sseq2d 3964	. . . . . . . . 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 4143	. . . . . . . . . . 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 4144	. . . . . . . . . . 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 4763	. . . . . . . . . . 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 4142	. . . . . . . . . 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 4142	. . . . . . . . 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 3577	. . . . . . 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 6831	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (𝑢‘0) = ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0))
56	55	eqeq1d 2736	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → ((𝑢‘0) = ℚ ↔ ((𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)‘0) = ℚ))
57		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑢 = (𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩) → (lastS‘𝑢) = (lastS‘(𝑣 ++ ⟨“(ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))”⟩)))
58	57	sseq2d 3964	. . . . . . . . 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 21311	. . . . . . . . . 10 ⊢ ℂ = (Base‘ℂfld)
61		cndrng 21351	. . . . . . . . . . 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 18529	. . . . . . . . . . . . . 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 6836	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ((𝐶‘𝑁) ∪ 𝑔) ⊆ (lastS‘𝑣)) ∧ 𝑣 = ∅) → (lastS‘𝑣) = (lastS‘∅))
66		lsw0g 14487	. . . . . . . . . . . . . . . . . . 19 ⊢ (lastS‘∅) = ∅
67	65, 66	eqtrdi 2785	. . . . . . . . . . . . . . . . . 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 4128	. . . . . . . . . . . . . . . . . . . . . 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 7812	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ On)
74	70, 73	constr01 33848	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → {0, 1} ⊆ (𝐶‘𝑁))
75		c0ex 11124	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
76	75	prnz 4732	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {0, 1} ≠ ∅
77		ssn0 4354	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (({0, 1} ⊆ (𝐶‘𝑁) ∧ {0, 1} ≠ ∅) → (𝐶‘𝑁) ≠ ∅)
78	74, 76, 77	sylancl 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝐶‘𝑁) ≠ ∅)
79		ssn0 4354	. . . . . . . . . . . . . . . . . . . . . 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 4354	. . . . . . . . . . . . . . . . . . . 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 2935	. . . . . . . . . . . . . . . . . 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 2937	. . . . . . . . . . . . . . . 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 32475	. . . . . . . . . . . . . 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 14489	. . . . . . . . . . . . . 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 20724	. . . . . . . . . . . 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 7753	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ On → suc 𝑁 ∈ On)
95	73, 94	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → suc 𝑁 ∈ On)
96	70, 95	constrsscn 33846	. . . . . . . . . . . . . 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 3911	. . . . . . . . . . . . . 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 3932	. . . . . . . . . . . 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 4763	. . . . . . . . . . 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 4142	. . . . . . . . . 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 33345	. . . . . . . . 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 3462	. . . . . . . . . . 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 3462	. . . . . . . . . . 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 2734	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (lastS‘𝑣)) = (ℂfld ↾s (lastS‘𝑣))
109		eqid 2734	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦}))) = (ℂfld ↾s (ℂfld fldGen ((lastS‘𝑣) ∪ {𝑦})))
110		cnfldfld 33372	. . . . . . . . . . . . 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 33774	. . . . . . . . . . 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 5105	. . . . . . . . . . . . . . 15 ⊢ (𝐸/FldExt𝐹 ↔ (ℂfld ↾s 𝑒)/FldExt(ℂfld ↾s 𝑓))
117		oveq2 7364	. . . . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . 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 5109	. . . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝐸[:]𝐹) = ((ℂfld ↾s 𝑒)[:](ℂfld ↾s 𝑓))
124	118, 120	oveq12d 7374	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 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 5480	. . . . . . . . . . . 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 18546	. . . . . . . 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 14525	. . . . . . . . . . 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 14309	. . . . . . . . . . . . 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 14505	. . . . . . . . . . 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 2769	. . . . . . . . 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 4130	. . . . . . . . . . . . 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 4144	. . . . . . . . . . . . . 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 4130	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 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 4142	. . . . . . . . . . . 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 4142	. . . . . . . . . . 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 33344	. . . . . . . . . . 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 3942	. . . . . . . . . 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 14556	. . . . . . . . . . 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 3969	. . . . . . . . 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 3577	. . . . . . 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 33853	. . . . . . 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 3137	. . . 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 7830	. . . . 5 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
168	71, 167	syl 17	. . . 4 ⊢ (𝜑 → suc 𝑁 ∈ ω)
169	70, 168	constrfin 33852	. . 3 ⊢ (𝜑 → (𝐶‘suc 𝑁) ∈ Fin)
170	4, 8, 19, 23, 35, 166, 169	findcard2d 9089	. 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 4144	. . . . 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 3147	. 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 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  {crab 3397  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  {csn 4578  {cpr 4580   class class class wbr 5096  {copab 5158   ↦ cmpt 5177  Oncon0 6315  suc csuc 6317  ‘cfv 6490  (class class class)co 7356  ωcom 7806  reccrdg 8338  ℂcc 11022  ℝcr 11023  0cc0 11024  1c1 11025   + caddc 11027   · cmul 11029   < clt 11164   − cmin 11362  2c2 12198  ℚcq 12859  ♯chash 14251  Word cword 14434  lastSclsw 14483   ++ cconcat 14491  ⟨“cs1 14517  ∗ccj 15017  ℑcim 15019  abscabs 15155   ↾_s cress 17155   Chain cchn 18526  DivRingcdr 20660  Fieldcfield 20661  SubDRingcsdrg 20717  ℂ_fldccnfld 21307   fldGen cfldgen 33341  /_FldExtcfldext 33744  [:]cextdg 33746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-reg 9495  ax-inf2 9548  ax-ac2 10371  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102  ax-addf 11103  ax-mulf 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-ofr 7621  df-rpss 7666  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-tpos 8166  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-ec 8635  df-qs 8639  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-sup 9343  df-inf 9344  df-oi 9413  df-r1 9674  df-rank 9675  df-dju 9811  df-card 9849  df-acn 9852  df-ac 10024  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-xnn0 12473  df-z 12487  df-dec 12606  df-uz 12750  df-rp 12904  df-ico 13265  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-word 14435  df-lsw 14484  df-concat 14492  df-s1 14518  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-starv 17190  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ocomp 17196  df-ds 17197  df-unif 17198  df-hom 17199  df-cco 17200  df-0g 17359  df-gsum 17360  df-prds 17365  df-pws 17367  df-imas 17427  df-qus 17428  df-mre 17503  df-mrc 17504  df-mri 17505  df-acs 17506  df-proset 18215  df-drs 18216  df-poset 18234  df-ipo 18449  df-chn 18527  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-submnd 18707  df-grp 18864  df-minusg 18865  df-sbg 18866  df-mulg 18996  df-subg 19051  df-nsg 19052  df-eqg 19053  df-ghm 19140  df-gim 19186  df-cntz 19244  df-oppg 19273  df-lsm 19563  df-cmn 19709  df-abl 19710  df-mgp 20074  df-rng 20086  df-ur 20115  df-srg 20120  df-ring 20168  df-cring 20169  df-oppr 20271  df-dvdsr 20291  df-unit 20292  df-irred 20293  df-invr 20322  df-dvr 20335  df-rhm 20406  df-nzr 20444  df-subrng 20477  df-subrg 20501  df-rlreg 20625  df-domn 20626  df-idom 20627  df-drng 20662  df-field 20663  df-sdrg 20718  df-lmod 20811  df-lss 20881  df-lsp 20921  df-lmhm 20972  df-lmim 20973  df-lmic 20974  df-lbs 21025  df-lvec 21053  df-sra 21123  df-rgmod 21124  df-lidl 21161  df-rsp 21162  df-2idl 21203  df-lpidl 21275  df-lpir 21276  df-pid 21290  df-cnfld 21308  df-dsmm 21685  df-frlm 21700  df-uvc 21736  df-lindf 21759  df-linds 21760  df-assa 21806  df-asp 21807  df-ascl 21808  df-psr 21863  df-mvr 21864  df-mpl 21865  df-opsr 21867  df-evls 22027  df-evl 22028  df-psr1 22118  df-vr1 22119  df-ply1 22120  df-coe1 22121  df-evls1 22257  df-evl1 22258  df-mdeg 26014  df-deg1 26015  df-mon1 26090  df-uc1p 26091  df-q1p 26092  df-r1p 26093  df-ig1p 26094  df-fldgen 33342  df-mxidl 33490  df-dim 33705  df-fldext 33747  df-extdg 33748  df-irng 33790  df-minply 33806

This theorem is referenced by:  constrextdg2  33855