Theorem constrmon 33726

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 2822	. . . 4 ⊢ (𝑚 = ∅ → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ ∅))
4		fveq2 6887	. . . . 5 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
5	4	sseq2d 3998	. . . 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 2817	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑛))
8		fveq2 6887	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
9	8	sseq2d 3998	. . . 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 2822	. . . 4 ⊢ (𝑚 = suc 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ suc 𝑛))
12		fveq2 6887	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
13	12	sseq2d 3998	. . . 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 2822	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑁))
16		fveq2 6887	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
17	16	sseq2d 3998	. . . 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 4320	. . . 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 33725	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
26	22, 25	sstrd 3976	. . . . 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 6891	. . . . . 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 33725	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
31	28, 30	eqsstrd 4000	. . . . 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 6432	. . . . . 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 6887	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝐶‘𝑖) = (𝐶‘𝑀))
38	37	sseq2d 3998	. . . . . . 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 3989	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑀))
41	38, 39, 40	rspcedvdw 3609	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → ∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖))
42		ssiun 5028	. . . . . 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 3468	. . . . . . 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 33721	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑚) = ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
48	43, 47	sseqtrrd 4003	. . . 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 7864	. 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 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  {crab 3420  Vcvv 3464   ⊆ wss 3933  ∅c0 4315  {cpr 4610  ∪ ciun 4973   ↦ cmpt 5207  Oncon0 6365  Lim wlim 6366  suc csuc 6367  ‘cfv 6542  (class class class)co 7414  reccrdg 8432  ℂcc 11136  ℝcr 11137  0cc0 11138  1c1 11139   + caddc 11141   · cmul 11143   − cmin 11475  ∗ccj 15118  ℑcim 15120  abscabs 15256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7871  df-2nd 7998  df-frecs 8289  df-wrecs 8320  df-recs 8394  df-rdg 8433  df-er 8728  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11280  df-mnf 11281  df-ltxr 11283  df-sub 11477

This theorem is referenced by: (None)