Theorem constrmon 33757

Description: The construction of constructible numbers is monotonous, i.e. if the ordinal 𝑀 is less than the ordinal 𝑁, which is denoted by 𝑀 ∈ 𝑁, then the 𝑀-th step of the constructible numbers is included in the 𝑁-th step. (Contributed by Thierry Arnoux, 1-Jul-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})
constrsscn.1	⊢ (𝜑 → 𝑁 ∈ On)
constrmon.1	⊢ (𝜑 → 𝑀 ∈ 𝑁)

Assertion

Ref	Expression
constrmon	⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))

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

Allowed substitution hints:   𝜑(𝑟,𝑑)   𝐶(𝑟)   𝑀(𝑟,𝑑)   𝑁(𝑟)

Proof of Theorem constrmon

Dummy variables 𝑛 𝑚 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		constrsscn.1	. 2 ⊢ (𝜑 → 𝑁 ∈ On)
2		constrmon.1	. 2 ⊢ (𝜑 → 𝑀 ∈ 𝑁)
3		eleq2 2820	. . . 4 ⊢ (𝑚 = ∅ → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ ∅))
4		fveq2 6822	. . . . 5 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
5	4	sseq2d 3962	. . . 4 ⊢ (𝑚 = ∅ → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘∅)))
6	3, 5	imbi12d 344	. . 3 ⊢ (𝑚 = ∅ → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))))
7		eleq2w 2815	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑛))
8		fveq2 6822	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
9	8	sseq2d 3962	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
10	7, 9	imbi12d 344	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))))
11		eleq2 2820	. . . 4 ⊢ (𝑚 = suc 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ suc 𝑛))
12		fveq2 6822	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
13	12	sseq2d 3962	. . . 4 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛)))
14	11, 13	imbi12d 344	. . 3 ⊢ (𝑚 = suc 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
15		eleq2 2820	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑁))
16		fveq2 6822	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
17	16	sseq2d 3962	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
18	15, 17	imbi12d 344	. . 3 ⊢ (𝑚 = 𝑁 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))))
19		noel 4285	. . . 4 ⊢ ¬ 𝑀 ∈ ∅
20	19	pm2.21i 119	. . 3 ⊢ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))
21		simpllr 775	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
22	21	syldbl2 841	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))
23		constr0.1	. . . . . . 7 ⊢ 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})
24		simplll 774	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → 𝑛 ∈ On)
25	23, 24	constrss 33756	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
26	22, 25	sstrd 3940	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
27		simpr 484	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑀 = 𝑛)
28	27	fveq2d 6826	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) = (𝐶‘𝑛))
29		simplll 774	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑛 ∈ On)
30	23, 29	constrss 33756	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
31	28, 30	eqsstrd 3964	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
32		simpr 484	. . . . . 6 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → 𝑀 ∈ suc 𝑛)
33		elsuci 6375	. . . . . 6 ⊢ (𝑀 ∈ suc 𝑛 → (𝑀 ∈ 𝑛 ∨ 𝑀 = 𝑛))
34	32, 33	syl 17	. . . . 5 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → (𝑀 ∈ 𝑛 ∨ 𝑀 = 𝑛))
35	26, 31, 34	mpjaodan 960	. . . 4 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
36	35	exp31 419	. . 3 ⊢ (𝑛 ∈ On → ((𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)) → (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
37		fveq2 6822	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝐶‘𝑖) = (𝐶‘𝑀))
38	37	sseq2d 3962	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑖) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑀)))
39		simpr 484	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑀 ∈ 𝑚)
40		ssidd 3953	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑀))
41	38, 39, 40	rspcedvdw 3575	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → ∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖))
42		ssiun 4993	. . . . . 6 ⊢ (∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖) → (𝐶‘𝑀) ⊆ ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
43	41, 42	syl 17	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
44		vex 3440	. . . . . . 7 ⊢ 𝑚 ∈ V
45	44	a1i 11	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑚 ∈ V)
46		simpll 766	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → Lim 𝑚)
47	23, 45, 46	constrlim 33752	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑚) = ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
48	43, 47	sseqtrrd 3967	. . . 4 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑚))
49	48	exp31 419	. . 3 ⊢ (Lim 𝑚 → (∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)) → (𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚))))
50	6, 10, 14, 18, 20, 36, 49	tfinds 7790	. 2 ⊢ (𝑁 ∈ On → (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
51	1, 2, 50	sylc 65	1 ⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ⊆ wss 3897  ∅c0 4280  {cpr 4575  ∪ ciun 4939   ↦ cmpt 5170  Oncon0 6306  Lim wlim 6307  suc csuc 6308  ‘cfv 6481  (class class class)co 7346  reccrdg 8328  ℂcc 11004  ℝcr 11005  0cc0 11006  1c1 11007   + caddc 11009   · cmul 11011   − cmin 11344  ∗ccj 15003  ℑcim 15005  abscabs 15141

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-ltxr 11151  df-sub 11346

This theorem is referenced by:  constrfiss  33764