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Theorem naddwordnexlem1 42750
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 (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
naddwordnexlem1 (𝜑𝐴𝐵)

Proof of Theorem naddwordnexlem1
StepHypRef Expression
1 naddwordnex.a . . 3 (𝜑𝐴 = ((ω ·o 𝐶) +o 𝑀))
2 naddwordnex.b . . 3 (𝜑𝐵 = ((ω ·o 𝐷) +o 𝑁))
3 naddwordnex.c . . 3 (𝜑𝐶𝐷)
4 naddwordnex.d . . 3 (𝜑𝐷 ∈ On)
5 naddwordnex.m . . 3 (𝜑𝑀 ∈ ω)
6 naddwordnex.n . . 3 (𝜑𝑁𝑀)
71, 2, 3, 4, 5, 6naddwordnexlem0 42749 . 2 (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
8 omelon 9661 . . . . . . 7 ω ∈ On
9 onelon 6388 . . . . . . . . 9 ((𝐷 ∈ On ∧ 𝐶𝐷) → 𝐶 ∈ On)
104, 3, 9syl2anc 583 . . . . . . . 8 (𝜑𝐶 ∈ On)
11 onsuc 7808 . . . . . . . 8 (𝐶 ∈ On → suc 𝐶 ∈ On)
1210, 11syl 17 . . . . . . 7 (𝜑 → suc 𝐶 ∈ On)
13 omcl 8550 . . . . . . 7 ((ω ∈ On ∧ suc 𝐶 ∈ On) → (ω ·o suc 𝐶) ∈ On)
148, 12, 13sylancr 586 . . . . . 6 (𝜑 → (ω ·o suc 𝐶) ∈ On)
15 onelss 6405 . . . . . 6 ((ω ·o suc 𝐶) ∈ On → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
1614, 15syl 17 . . . . 5 (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
1716adantrd 491 . . . 4 (𝜑 → ((𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵) → 𝐴 ⊆ (ω ·o suc 𝐶)))
1817imp 406 . . 3 ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ (ω ·o suc 𝐶))
19 simprr 772 . . 3 ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → (ω ·o suc 𝐶) ⊆ 𝐵)
2018, 19sstrd 3988 . 2 ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴𝐵)
217, 20mpdan 686 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wss 3944  Oncon0 6363  suc csuc 6365  (class class class)co 7414  ωcom 7864   +o coa 8477   ·o comu 8478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-oadd 8484  df-omul 8485
This theorem is referenced by:  oawordex3  42753
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