Theorem naddwordnexlem0 43848

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 9565	. . . . . 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 6342	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
6	3, 4, 5	syl2anc 590	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
7		omcl 8468	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
8	2, 6, 7	syl2anc 590	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
9	2, 8	jca 516	. . . 4 ⊢ (𝜑 → (ω ∈ On ∧ (ω ·o 𝐶) ∈ On))
10		naddwordnex.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ω)
11		oaordi 8478	. . . 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 8458	. . . 4 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
15	2, 6, 14	syl2anc 590	. . 3 ⊢ (𝜑 → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
16	12, 13, 15	3eltr4d 2855	. 2 ⊢ (𝜑 → 𝐴 ∈ (ω ·o suc 𝐶))
17		onsuc 7760	. . . . . . 7 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
18	6, 17	syl 17	. . . . . 6 ⊢ (𝜑 → suc 𝐶 ∈ On)
19	18, 3, 2	3jca 1134	. . . . 5 ⊢ (𝜑 → (suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On))
20		onsucss 43718	. . . . . 6 ⊢ (𝐷 ∈ On → (𝐶 ∈ 𝐷 → suc 𝐶 ⊆ 𝐷))
21	3, 4, 20	sylc 65	. . . . 5 ⊢ (𝜑 → suc 𝐶 ⊆ 𝐷)
22		omwordi 8503	. . . . 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 8468	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
25	2, 3, 24	syl2anc 590	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On)
26		naddwordnex.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑀)
27	26, 10	jca 516	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
28		ontr1 6364	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
29	2, 27, 28	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
30		nnon 7819	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
32		oaword1 8484	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
33	25, 31, 32	syl2anc 590	. . . 4 ⊢ (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
34	23, 33	sstrd 3932	. . 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 3959	. 2 ⊢ (𝜑 → (ω ·o suc 𝐶) ⊆ 𝐵)
37	16, 36	jca 516	1 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  Oncon0 6317  suc csuc 6319  (class class class)co 7363  ωcom 7813   +_o coa 8399   ·_o comu 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-inf2 9560

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-oadd 8406  df-omul 8407

This theorem is referenced by:  naddwordnexlem1  43849  naddwordnexlem2  43850  naddwordnexlem3  43851