Theorem constrmon 34134

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 2852	. . . 4 ⊢ (𝑚 = ∅ → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ ∅))
4		fveq2 6881	. . . . 5 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
5	4	sseq2d 3969	. . . 4 ⊢ (𝑚 = ∅ → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘∅)))
6	3, 5	imbi12d 347	. . 3 ⊢ (𝑚 = ∅ → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))))
7		eleq2w 2847	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑛))
8		fveq2 6881	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
9	8	sseq2d 3969	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
10	7, 9	imbi12d 347	. . 3 ⊢ (𝑚 = 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))))
11		eleq2 2852	. . . 4 ⊢ (𝑚 = suc 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ suc 𝑛))
12		fveq2 6881	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
13	12	sseq2d 3969	. . . 4 ⊢ (𝑚 = suc 𝑛 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛)))
14	11, 13	imbi12d 347	. . 3 ⊢ (𝑚 = suc 𝑛 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
15		eleq2 2852	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑁))
16		fveq2 6881	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
17	16	sseq2d 3969	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑚) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
18	15, 17	imbi12d 347	. . 3 ⊢ (𝑚 = 𝑁 → ((𝑀 ∈ 𝑚 → (𝐶‘𝑀) ⊆ (𝐶‘𝑚)) ↔ (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))))
19		noel 4291	. . . 4 ⊢ ¬ 𝑀 ∈ ∅
20	19	pm2.21i 120	. . 3 ⊢ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))
21		simpllr 787	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
22	21	syldbl2 854	. . . . . 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 786	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → 𝑛 ∈ On)
25	23, 24	constrss 34133	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
26	22, 25	sstrd 3947	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
27		simpr 489	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑀 = 𝑛)
28	27	fveq2d 6885	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) = (𝐶‘𝑛))
29		simplll 786	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑛 ∈ On)
30	23, 29	constrss 34133	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
31	28, 30	eqsstrd 3971	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
32		simpr 489	. . . . . 6 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → 𝑀 ∈ suc 𝑛)
33		elsuci 6430	. . . . . 6 ⊢ (𝑀 ∈ suc 𝑛 → (𝑀 ∈ 𝑛 ∨ 𝑀 = 𝑛))
34	32, 33	syl 18	. . . . 5 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → (𝑀 ∈ 𝑛 ∨ 𝑀 = 𝑛))
35	26, 31, 34	mpjaodan 973	. . . 4 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
36	35	exp31 424	. . 3 ⊢ (𝑛 ∈ On → ((𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)) → (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
37		fveq2 6881	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝐶‘𝑖) = (𝐶‘𝑀))
38	37	sseq2d 3969	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑖) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑀)))
39		simpr 489	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑀 ∈ 𝑚)
40		ssidd 3960	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑀))
41	38, 39, 40	rspcedvdw 3584	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → ∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖))
42		ssiun 5011	. . . . . 6 ⊢ (∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖) → (𝐶‘𝑀) ⊆ ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
43	41, 42	syl 18	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
44		vex 3459	. . . . . . 7 ⊢ 𝑚 ∈ V
45	44	a1i 11	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑚 ∈ V)
46		simpll 778	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → Lim 𝑚)
47	23, 45, 46	constrlim 34129	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑚) = ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
48	43, 47	sseqtrrd 3974	. . . 4 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑚))
49	48	exp31 424	. . 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 7852	. 2 ⊢ (𝑁 ∈ On → (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
51	1, 2, 50	sylc 66	1 ⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  {cpr 4591  ∪ ciun 4956   ↦ cmpt 5192  Oncon0 6360  Lim wlim 6361  suc csuc 6362  ‘cfv 6536  (class class class)co 7410  reccrdg 8392  ℂcc 11093  ℝcr 11094  0cc0 11095  1c1 11096   + caddc 11098   · cmul 11100   − cmin 11436  ∗ccj 15143  ℑcim 15145  abscabs 15281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-ltxr 11243  df-sub 11438

This theorem is referenced by:  constrfiss  34141