Theorem naddwordnexlem1 43388

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, 𝐵 is equal to or larger than 𝐴. (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
naddwordnexlem1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem naddwordnexlem1

Step	Hyp	Ref	Expression
1		naddwordnex.a	. . 3 ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀))
2		naddwordnex.b	. . 3 ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁))
3		naddwordnex.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		naddwordnex.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ On)
5		naddwordnex.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ω)
6		naddwordnex.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑀)
7	1, 2, 3, 4, 5, 6	naddwordnexlem0 43387	. 2 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
8		omelon 9682	. . . . . . 7 ⊢ ω ∈ On
9		onelon 6407	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
10	4, 3, 9	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
11		onsuc 7827	. . . . . . . 8 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → suc 𝐶 ∈ On)
13		omcl 8570	. . . . . . 7 ⊢ ((ω ∈ On ∧ suc 𝐶 ∈ On) → (ω ·o suc 𝐶) ∈ On)
14	8, 12, 13	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ω ·o suc 𝐶) ∈ On)
15		onelss 6424	. . . . . 6 ⊢ ((ω ·o suc 𝐶) ∈ On → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
17	16	adantrd 491	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵) → 𝐴 ⊆ (ω ·o suc 𝐶)))
18	17	imp 406	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ (ω ·o suc 𝐶))
19		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → (ω ·o suc 𝐶) ⊆ 𝐵)
20	18, 19	sstrd 3993	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
21	7, 20	mpdan 687	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ⊆ wss 3950  Oncon0 6382  suc csuc 6384  (class class class)co 7429  ωcom 7883   +_o coa 8499   ·_o comu 8500

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751  ax-inf2 9677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-tr 5258  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-ord 6385  df-on 6386  df-lim 6387  df-suc 6388  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-om 7884  df-2nd 8011  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-oadd 8506  df-omul 8507

This theorem is referenced by:  oawordex3  43391