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Theorem naddwordnexlem1 43345
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 43344 . 2 (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
8 omelon 9677 . . . . . . 7 ω ∈ On
9 onelon 6405 . . . . . . . . 9 ((𝐷 ∈ On ∧ 𝐶𝐷) → 𝐶 ∈ On)
104, 3, 9syl2anc 583 . . . . . . . 8 (𝜑𝐶 ∈ On)
11 onsuc 7824 . . . . . . . 8 (𝐶 ∈ On → suc 𝐶 ∈ On)
1210, 11syl 17 . . . . . . 7 (𝜑 → suc 𝐶 ∈ On)
13 omcl 8567 . . . . . . 7 ((ω ∈ On ∧ suc 𝐶 ∈ On) → (ω ·o suc 𝐶) ∈ On)
148, 12, 13sylancr 586 . . . . . 6 (𝜑 → (ω ·o suc 𝐶) ∈ On)
15 onelss 6422 . . . . . 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 4006 . 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 1535  wcel 2104  wss 3963  Oncon0 6380  suc csuc 6382  (class class class)co 7425  ωcom 7880   +o coa 8496   ·o comu 8497
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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5430  ax-un 7747  ax-inf2 9672
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6317  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-om 7881  df-2nd 8008  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-oadd 8503  df-omul 8504
This theorem is referenced by:  oawordex3  43348
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