Theorem naddwordnexlem0 42610

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 9647	. . . . . 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 6389	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
6	3, 4, 5	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
7		omcl 8542	. . . . . 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 8552	. . . 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 8532	. . . 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 2847	. 2 ⊢ (𝜑 → 𝐴 ∈ (ω ·o suc 𝐶))
17		onsuc 7803	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
18	6, 17	syl 17	. . . . . 6 ⊢ (𝜑 → suc 𝐶 ∈ On)
19	18, 3, 2	3jca 1127	. . . . 5 ⊢ (𝜑 → (suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On))
20		onsucss 42479	. . . . . 6 ⊢ (𝐷 ∈ On → (𝐶 ∈ 𝐷 → suc 𝐶 ⊆ 𝐷))
21	3, 4, 20	sylc 65	. . . . 5 ⊢ (𝜑 → suc 𝐶 ⊆ 𝐷)
22		omwordi 8577	. . . . 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 8542	. . . . . 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 6410	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
29	2, 27, 28	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nnon 7865	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
32		oaword1 8558	. . . . 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 3992	. . 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 4023	. 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 2105   ⊆ wss 3948  Oncon0 6364  suc csuc 6366  (class class class)co 7412  ωcom 7859   +_o coa 8469   ·_o comu 8470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729  ax-inf2 9642

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476  df-omul 8477

This theorem is referenced by:  naddwordnexlem1  42611  naddwordnexlem2  42612  naddwordnexlem3  42613