MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oaabs Structured version   Visualization version   GIF version

Theorem oaabs 7893
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 4956 . . . . . . . . 9 ((ω ⊆ 𝐵𝐵 ∈ On) → ω ∈ V)
21ex 449 . . . . . . . 8 (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3 omelon2 7242 . . . . . . . 8 (ω ∈ V → ω ∈ On)
42, 3syl6com 37 . . . . . . 7 (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
54imp 444 . . . . . 6 ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
65adantll 752 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7 simplr 809 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
86, 7jca 555 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9 oawordeu 7804 . . . 4 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
108, 9sylancom 704 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
11 reurex 3299 . . 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 7236 . . . . . . 7 (𝐴 ∈ ω → 𝐴 ∈ On)
1413ad3antrrr 768 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
156adantr 472 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16 simpr 479 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17 oaass 7810 . . . . . 6 ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
1814, 15, 16, 17syl3anc 1477 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
19 simpll 807 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20 oaabslem 7892 . . . . . . . 8 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω)
216, 19, 20syl2anc 696 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +𝑜 ω) = ω)
2221adantr 472 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 ω) = ω)
2322oveq1d 6828 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (ω +𝑜 𝑥))
2418, 23eqtr3d 2796 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥))
25 oveq2 6821 . . . . 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 2775 . . . 4 ((ω +𝑜 𝑥) = 𝐵 → ((𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) = 𝐵))
2824, 27syl5ibcom 235 . . 3 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +𝑜 𝑥) = 𝐵 → (𝐴 +𝑜 𝐵) = 𝐵))
2928rexlimdva 3169 . 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 383   = wceq 1632  wcel 2139  wrex 3051  ∃!wreu 3052  Vcvv 3340  wss 3715  Oncon0 5884  (class class class)co 6813  ωcom 7230   +𝑜 coa 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator