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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 (𝜑𝐴 = ((ω ·o 𝐶) +o 𝑀))
naddwordnex.b (𝜑𝐵 = ((ω ·o 𝐷) +o 𝑁))
naddwordnex.c (𝜑𝐶𝐷)
naddwordnex.d (𝜑𝐷 ∈ On)
naddwordnex.m (𝜑𝑀 ∈ ω)
naddwordnex.n (𝜑𝑁𝑀)
Assertion
Ref Expression
naddwordnexlem0 (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))

Proof of Theorem naddwordnexlem0
StepHypRef Expression
1 omelon 9647 . . . . . 6 ω ∈ On
21a1i 11 . . . . 5 (𝜑 → ω ∈ On)
3 naddwordnex.d . . . . . . 7 (𝜑𝐷 ∈ On)
4 naddwordnex.c . . . . . . 7 (𝜑𝐶𝐷)
5 onelon 6389 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐶𝐷) → 𝐶 ∈ On)
63, 4, 5syl2anc 583 . . . . . 6 (𝜑𝐶 ∈ On)
7 omcl 8542 . . . . . 6 ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
82, 6, 7syl2anc 583 . . . . 5 (𝜑 → (ω ·o 𝐶) ∈ On)
92, 8jca 511 . . . 4 (𝜑 → (ω ∈ On ∧ (ω ·o 𝐶) ∈ On))
10 naddwordnex.m . . . 4 (𝜑𝑀 ∈ ω)
11 oaordi 8552 . . . 4 ((ω ∈ On ∧ (ω ·o 𝐶) ∈ On) → (𝑀 ∈ ω → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω)))
129, 10, 11sylc 65 . . 3 (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ ((ω ·o 𝐶) +o ω))
13 naddwordnex.a . . 3 (𝜑𝐴 = ((ω ·o 𝐶) +o 𝑀))
14 omsuc 8532 . . . 4 ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
152, 6, 14syl2anc 583 . . 3 (𝜑 → (ω ·o suc 𝐶) = ((ω ·o 𝐶) +o ω))
1612, 13, 153eltr4d 2847 . 2 (𝜑𝐴 ∈ (ω ·o suc 𝐶))
17 onsuc 7803 . . . . . . 7 (𝐶 ∈ On → suc 𝐶 ∈ On)
186, 17syl 17 . . . . . 6 (𝜑 → suc 𝐶 ∈ On)
1918, 3, 23jca 1127 . . . . 5 (𝜑 → (suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On))
20 onsucss 42479 . . . . . 6 (𝐷 ∈ On → (𝐶𝐷 → suc 𝐶𝐷))
213, 4, 20sylc 65 . . . . 5 (𝜑 → suc 𝐶𝐷)
22 omwordi 8577 . . . . 5 ((suc 𝐶 ∈ On ∧ 𝐷 ∈ On ∧ ω ∈ On) → (suc 𝐶𝐷 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷)))
2319, 21, 22sylc 65 . . . 4 (𝜑 → (ω ·o suc 𝐶) ⊆ (ω ·o 𝐷))
24 omcl 8542 . . . . . 6 ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
252, 3, 24syl2anc 583 . . . . 5 (𝜑 → (ω ·o 𝐷) ∈ On)
26 naddwordnex.n . . . . . . . 8 (𝜑𝑁𝑀)
2726, 10jca 511 . . . . . . 7 (𝜑 → (𝑁𝑀𝑀 ∈ ω))
28 ontr1 6410 . . . . . . 7 (ω ∈ On → ((𝑁𝑀𝑀 ∈ ω) → 𝑁 ∈ ω))
292, 27, 28sylc 65 . . . . . 6 (𝜑𝑁 ∈ ω)
30 nnon 7865 . . . . . 6 (𝑁 ∈ ω → 𝑁 ∈ On)
3129, 30syl 17 . . . . 5 (𝜑𝑁 ∈ On)
32 oaword1 8558 . . . . 5 (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
3325, 31, 32syl2anc 583 . . . 4 (𝜑 → (ω ·o 𝐷) ⊆ ((ω ·o 𝐷) +o 𝑁))
3423, 33sstrd 3992 . . 3 (𝜑 → (ω ·o suc 𝐶) ⊆ ((ω ·o 𝐷) +o 𝑁))
35 naddwordnex.b . . 3 (𝜑𝐵 = ((ω ·o 𝐷) +o 𝑁))
3634, 35sseqtrrd 4023 . 2 (𝜑 → (ω ·o suc 𝐶) ⊆ 𝐵)
3716, 36jca 511 1 (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
Colors of variables: wff setvar class
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
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