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Theorem oaabs 8260
Description: Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
oaabs (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)

Proof of Theorem oaabs
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssexg 5218 . . . . . . . . 9 ((ω ⊆ 𝐵𝐵 ∈ On) → ω ∈ V)
21ex 413 . . . . . . . 8 (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3 omelon2 7581 . . . . . . . 8 (ω ∈ V → ω ∈ On)
42, 3syl6com 37 . . . . . . 7 (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
54imp 407 . . . . . 6 ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
65adantll 710 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7 simplr 765 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
86, 7jca 512 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9 oawordeu 8170 . . . 4 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
108, 9sylancom 588 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
11 reurex 3429 . . 3 (∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
1210, 11syl 17 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
13 nnon 7575 . . . . . . 7 (𝐴 ∈ ω → 𝐴 ∈ On)
1413ad3antrrr 726 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
156adantr 481 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16 simpr 485 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17 oaass 8176 . . . . . 6 ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
1814, 15, 16, 17syl3anc 1363 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
19 simpll 763 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20 oaabslem 8259 . . . . . . . 8 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
216, 19, 20syl2anc 584 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o ω) = ω)
2221adantr 481 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o ω) = ω)
2322oveq1d 7160 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (ω +o 𝑥))
2418, 23eqtr3d 2855 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥))
25 oveq2 7153 . . . . 5 ((ω +o 𝑥) = 𝐵 → (𝐴 +o (ω +o 𝑥)) = (𝐴 +o 𝐵))
26 id 22 . . . . 5 ((ω +o 𝑥) = 𝐵 → (ω +o 𝑥) = 𝐵)
2725, 26eqeq12d 2834 . . . 4 ((ω +o 𝑥) = 𝐵 → ((𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
2824, 27syl5ibcom 246 . . 3 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
2928rexlimdva 3281 . 2 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (∃𝑥 ∈ On (ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
3012, 29mpd 15 1 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3136  ∃!wreu 3137  Vcvv 3492  wss 3933  Oncon0 6184  (class class class)co 7145  ωcom 7569   +o coa 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095
This theorem is referenced by: (None)
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