Theorem naddwordnexlem0 43360

Description: When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, (ω ·_o suc 𝐶) lies between 𝐴 and 𝐵. (Contributed by RP, 14-Feb-2025.)

Hypotheses

Ref	Expression
naddwordnex.a	⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))
naddwordnex.b	⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))
naddwordnex.c	⊢ (𝜑 → 𝐶 ∈ 𝐷)
naddwordnex.d	⊢ (𝜑 → 𝐷 ∈ On)
naddwordnex.m	⊢ (𝜑 → 𝑀 ∈ ω)
naddwordnex.n	⊢ (𝜑 → 𝑁 ∈ 𝑀)

Assertion

Ref	Expression
naddwordnexlem0	⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))

Proof of Theorem naddwordnexlem0

Step	Hyp	Ref	Expression
1		omelon 9717	. . . . . 6 ⊢ ω ∈ On
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → ω ∈ On)
3		naddwordnex.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ On)
4		naddwordnex.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
5		onelon 6422	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
6	3, 4, 5	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
7		omcl 8594	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
8	2, 6, 7	syl2anc 583	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
9	2, 8	jca 511	. . . 4 ⊢ (𝜑 → (ω ∈ On ∧ (ω ·o 𝐶) ∈ On))
10		naddwordnex.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ω)
11		oaordi 8604	. . . 4 ⊢ ((ω ∈ On ∧ (ω ·o 𝐶) ∈ On) → (𝑀 ∈ ω → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω)))
12	9, 10, 11	sylc 65	. . 3 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω))
13		naddwordnex.a	. . 3 ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))
14		omsuc 8584	. . . 4 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
15	2, 6, 14	syl2anc 583	. . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
16	12, 13, 15	3eltr4d 2859	. 2 ⊢ (𝜑 → 𝐴 ∈ (ω ·o suc 𝐶))
17		onsuc 7849	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
18	6, 17	syl 17	. . . . . 6 ⊢ (𝜑 → suc 𝐶 ∈ On)
19	18, 3, 2	3jca 1128	. . . . 5 ⊢ (𝜑 → (suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On))
20		onsucss 43230	. . . . . 6 ⊢ (𝐷 ∈ On → (𝐶 ∈ 𝐷 → suc 𝐶 ⊆ 𝐷))
21	3, 4, 20	sylc 65	. . . . 5 ⊢ (𝜑 → suc 𝐶 ⊆ 𝐷)
22		omwordi 8629	. . . . 5 ⊢ ((suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On) → (suc 𝐶 ⊆ 𝐷 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷)))
23	19, 21, 22	sylc 65	. . . 4 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷))
24		omcl 8594	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
25	2, 3, 24	syl2anc 583	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On)
26		naddwordnex.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑀)
27	26, 10	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
28		ontr1 6443	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
29	2, 27, 28	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nnon 7911	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
32		oaword1 8610	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
33	25, 31, 32	syl2anc 583	. . . 4 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
34	23, 33	sstrd 4019	. . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ ((ω ·o 𝐷) +o 𝑁))
35		naddwordnex.b	. . 3 ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))
36	34, 35	sseqtrrd 4050	. 2 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ 𝐵)
37	16, 36	jca 511	1 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  Oncon0 6397  suc csuc 6399  (class class class)co 7450  ωcom 7905   +_o coa 8521   ·_o comu 8522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7772  ax-inf2 9712

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-oadd 8528  df-omul 8529

This theorem is referenced by:  naddwordnexlem1  43361  naddwordnexlem2  43362  naddwordnexlem3  43363