Theorem constrmon 34257

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 2849	. . . 4 ⊢ (𝑚 = ∅ → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ ∅))
4		fveq2 6879	. . . . 5 ⊢ (𝑚 = ∅ → (𝐶‘𝑚) = (𝐶‘∅))
5	4	sseq2d 3963	. . . 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 2844	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑛))
8		fveq2 6879	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐶‘𝑚) = (𝐶‘𝑛))
9	8	sseq2d 3963	. . . 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 2849	. . . 4 ⊢ (𝑚 = suc 𝑛 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ suc 𝑛))
12		fveq2 6879	. . . . 5 ⊢ (𝑚 = suc 𝑛 → (𝐶‘𝑚) = (𝐶‘suc 𝑛))
13	12	sseq2d 3963	. . . 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 2849	. . . 4 ⊢ (𝑚 = 𝑁 → (𝑀 ∈ 𝑚 ↔ 𝑀 ∈ 𝑁))
16		fveq2 6879	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐶‘𝑚) = (𝐶‘𝑁))
17	16	sseq2d 3963	. . . 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 4284	. . . 4 ⊢ ¬ 𝑀 ∈ ∅
20	19	pm2.21i 120	. . 3 ⊢ (𝑀 ∈ ∅ → (𝐶‘𝑀) ⊆ (𝐶‘∅))
21		simpllr 788	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)))
22	21	syldbl2 855	. . . . . 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 787	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → 𝑛 ∈ On)
25	23, 24	constrss 34256	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
26	22, 25	sstrd 3941	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 ∈ 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
27		simpr 490	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑀 = 𝑛)
28	27	fveq2d 6883	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) = (𝐶‘𝑛))
29		simplll 787	. . . . . . 7 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → 𝑛 ∈ On)
30	23, 29	constrss 34256	. . . . . 6 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑛) ⊆ (𝐶‘suc 𝑛))
31	28, 30	eqsstrd 3965	. . . . 5 ⊢ ((((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) ∧ 𝑀 = 𝑛) → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))
32		simpr 490	. . . . . 6 ⊢ (((𝑛 ∈ On ∧ (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ suc 𝑛) → 𝑀 ∈ suc 𝑛)
33		elsuci 6427	. . . . . 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 425	. . 3 ⊢ (𝑛 ∈ On → ((𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛)) → (𝑀 ∈ suc 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘suc 𝑛))))
37		fveq2 6879	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝐶‘𝑖) = (𝐶‘𝑀))
38	37	sseq2d 3963	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝐶‘𝑀) ⊆ (𝐶‘𝑖) ↔ (𝐶‘𝑀) ⊆ (𝐶‘𝑀)))
39		simpr 490	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑀 ∈ 𝑚)
40		ssidd 3954	. . . . . . 7 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑀))
41	38, 39, 40	rspcedvdw 3579	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → ∃𝑖 ∈ 𝑚 (𝐶‘𝑀) ⊆ (𝐶‘𝑖))
42		ssiun 5005	. . . . . 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 3454	. . . . . . 7 ⊢ 𝑚 ∈ V
45	44	a1i 11	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → 𝑚 ∈ V)
46		simpll 779	. . . . . 6 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → Lim 𝑚)
47	23, 45, 46	constrlim 34252	. . . . 5 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑚) = ∪ 𝑖 ∈ 𝑚 (𝐶‘𝑖))
48	43, 47	sseqtrrd 3968	. . . 4 ⊢ (((Lim 𝑚 ∧ ∀𝑛 ∈ 𝑚 (𝑀 ∈ 𝑛 → (𝐶‘𝑀) ⊆ (𝐶‘𝑛))) ∧ 𝑀 ∈ 𝑚) → (𝐶‘𝑀) ⊆ (𝐶‘𝑚))
49	48	exp31 425	. . 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 7857	. 2 ⊢ (𝑁 ∈ On → (𝑀 ∈ 𝑁 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁)))
51	1, 2, 50	sylc 66	1 ⊢ (𝜑 → (𝐶‘𝑀) ⊆ (𝐶‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3899  ∅c0 4279  {cpr 4586  ∪ ciun 4951   ↦ cmpt 5186  Oncon0 6357  Lim wlim 6358  suc csuc 6359  ‘cfv 6533  (class class class)co 7414  reccrdg 8399  ℂcc 11125  ℝcr 11126  0cc0 11127  1c1 11128   + caddc 11130   · cmul 11132   − cmin 11468  ∗ccj 15186  ℑcim 15188  abscabs 15324

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-ltxr 11275  df-sub 11470

This theorem is used by:  constrfiss  34264