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Theorem oawordex3 42640
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, some ordinal sum of 𝐴 is equal to 𝐵. This is a specialization of oawordex 8552. (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
oawordex3 (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑀(𝑥)   𝑁(𝑥)

Proof of Theorem oawordex3
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, 6naddwordnexlem1 42637 . 2 (𝜑𝐴𝐵)
8 omelon 9637 . . . . . . 7 ω ∈ On
98a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
10 onelon 6379 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐶𝐷) → 𝐶 ∈ On)
114, 3, 10syl2anc 583 . . . . . 6 (𝜑𝐶 ∈ On)
12 omcl 8531 . . . . . 6 ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
139, 11, 12syl2anc 583 . . . . 5 (𝜑 → (ω ·o 𝐶) ∈ On)
14 nnon 7854 . . . . . 6 (𝑀 ∈ ω → 𝑀 ∈ On)
155, 14syl 17 . . . . 5 (𝜑𝑀 ∈ On)
16 oacl 8530 . . . . 5 (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On)
1713, 15, 16syl2anc 583 . . . 4 (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On)
181, 17eqeltrd 2825 . . 3 (𝜑𝐴 ∈ On)
19 omcl 8531 . . . . . 6 ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
209, 4, 19syl2anc 583 . . . . 5 (𝜑 → (ω ·o 𝐷) ∈ On)
216, 5jca 511 . . . . . . 7 (𝜑 → (𝑁𝑀𝑀 ∈ ω))
22 ontr1 6400 . . . . . . 7 (ω ∈ On → ((𝑁𝑀𝑀 ∈ ω) → 𝑁 ∈ ω))
239, 21, 22sylc 65 . . . . . 6 (𝜑𝑁 ∈ ω)
24 nnon 7854 . . . . . 6 (𝑁 ∈ ω → 𝑁 ∈ On)
2523, 24syl 17 . . . . 5 (𝜑𝑁 ∈ On)
26 oacl 8530 . . . . 5 (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On)
2720, 25, 26syl2anc 583 . . . 4 (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On)
282, 27eqeltrd 2825 . . 3 (𝜑𝐵 ∈ On)
29 oawordex 8552 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
3018, 28, 29syl2anc 583 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
317, 30mpbid 231 1 (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3062  wss 3940  Oncon0 6354  (class class class)co 7401  ωcom 7848   +o coa 8458   ·o comu 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-inf2 9632
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-oadd 8465  df-omul 8466
This theorem is referenced by: (None)
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