Theorem constrmon 33616

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 2815	. . . 4 ⊢ (𝑚 = ∅ → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ ∅))
4		fveq2 6893	. . . . 5 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
5	4	sseq2d 4011	. . . 4 ⊢ (𝑚 = ∅ → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘∅)))
6	3, 5	imbi12d 343	. . 3 ⊢ (𝑚 = ∅ → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))))
7		eleq2w 2810	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑛))
8		fveq2 6893	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
9	8	sseq2d 4011	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
10	7, 9	imbi12d 343	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))))
11		eleq2 2815	. . . 4 ⊢ (𝑚 = suc 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ suc 𝑛))
12		fveq2 6893	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
13	12	sseq2d 4011	. . . 4 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛)))
14	11, 13	imbi12d 343	. . 3 ⊢ (𝑚 = suc 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
15		eleq2 2815	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑁))
16		fveq2 6893	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
17	16	sseq2d 4011	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
18	15, 17	imbi12d 343	. . 3 ⊢ (𝑚 = 𝑁 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))))
19		noel 4330	. . . 4 ⊢ ¬ 𝑀 ∈ ∅
20	19	pm2.21i 119	. . 3 ⊢ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))
21		simpllr 774	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
22	21	syldbl2 839	. . . . . 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 773	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → 𝑛 ∈ On)
25	23, 24	constrss 33615	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
26	22, 25	sstrd 3989	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
27		simpr 483	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑀 = 𝑛)
28	27	fveq2d 6897	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) = (𝐶‘𝑛))
29		simplll 773	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑛 ∈ On)
30	23, 29	constrss 33615	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
31	28, 30	eqsstrd 4017	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
32		simpr 483	. . . . . 6 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → 𝑀 ∈ suc 𝑛)
33		elsuci 6435	. . . . . 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 956	. . . 4 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
36	35	exp31 418	. . 3 ⊢ (𝑛 ∈ On → ((𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)) → (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
37		fveq2 6893	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝐶‘𝑖) = (𝐶‘𝑀))
38	37	sseq2d 4011	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑖) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑀)))
39		simpr 483	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑀 ∈ 𝑚)
40		ssidd 4002	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑀))
41	38, 39, 40	rspcedvdw 3610	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → ∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖))
42		ssiun 5046	. . . . . 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 3466	. . . . . . 7 ⊢ 𝑚 ∈ V
45	44	a1i 11	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑚 ∈ V)
46		simpll 765	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → Lim 𝑚)
47	23, 45, 46	constrlim 33611	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑚) = ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
48	43, 47	sseqtrrd 4020	. . . 4 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑚))
49	48	exp31 418	. . 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 7862	. 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 394   ∨ wo 845   ∨ w3o 1083   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {crab 3419  Vcvv 3462   ⊆ wss 3946  ∅c0 4322  {cpr 4625  ∪ ciun 4993   ↦ cmpt 5228  Oncon0 6368  Lim wlim 6369  suc csuc 6370  ‘cfv 6546  (class class class)co 7416  reccrdg 8431  ℂcc 11147  ℝcr 11148  0cc0 11149  1c1 11150   + caddc 11152   · cmul 11154   − cmin 11485  ∗ccj 15096  ℑcim 15098  abscabs 15234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-ltxr 11294  df-sub 11487

This theorem is referenced by: (None)