Theorem naddwordnexlem0 43357

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 9617	. . . . . 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 6365	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
6	3, 4, 5	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
7		omcl 8511	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
8	2, 6, 7	syl2anc 584	. . . . 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 8521	. . . 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 8501	. . . 4 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
15	2, 6, 14	syl2anc 584	. . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
16	12, 13, 15	3eltr4d 2844	. 2 ⊢ (𝜑 → 𝐴 ∈ (ω ·o suc 𝐶))
17		onsuc 7794	. . . . . . 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 43227	. . . . . 6 ⊢ (𝐷 ∈ On → (𝐶 ∈ 𝐷 → suc 𝐶 ⊆ 𝐷))
21	3, 4, 20	sylc 65	. . . . 5 ⊢ (𝜑 → suc 𝐶 ⊆ 𝐷)
22		omwordi 8546	. . . . 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 8511	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
25	2, 3, 24	syl2anc 584	. . . . 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 6387	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
29	2, 27, 28	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nnon 7856	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
32		oaword1 8527	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
33	25, 31, 32	syl2anc 584	. . . 4 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
34	23, 33	sstrd 3965	. . 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 3992	. 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 1086   = wceq 1540   ∈ wcel 2109   ⊆ wss 3922  Oncon0 6340  suc csuc 6342  (class class class)co 7394  ωcom 7850   +_o coa 8440   ·_o comu 8441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718  ax-inf2 9612

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5541  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7851  df-2nd 7978  df-frecs 8269  df-wrecs 8300  df-recs 8349  df-rdg 8387  df-oadd 8447  df-omul 8448

This theorem is referenced by:  naddwordnexlem1  43358  naddwordnexlem2  43359  naddwordnexlem3  43360