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

Theorem oaabs 7588
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 4727 . . . . . . . . 9 ((ω ⊆ 𝐵𝐵 ∈ On) → ω ∈ V)
21ex 448 . . . . . . . 8 (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3 omelon2 6946 . . . . . . . 8 (ω ∈ V → ω ∈ On)
42, 3syl6com 36 . . . . . . 7 (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
54imp 443 . . . . . 6 ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
65adantll 745 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7 simplr 787 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
86, 7jca 552 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9 oawordeu 7499 . . . 4 (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
108, 9sylancom 697 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
11 reurex 3136 . . 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 6940 . . . . . . 7 (𝐴 ∈ ω → 𝐴 ∈ On)
1413ad3antrrr 761 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
156adantr 479 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16 simpr 475 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17 oaass 7505 . . . . . 6 ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
1814, 15, 16, 17syl3anc 1317 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
19 simpll 785 . . . . . . . 8 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20 oaabslem 7587 . . . . . . . 8 ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω)
216, 19, 20syl2anc 690 . . . . . . 7 (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +𝑜 ω) = ω)
2221adantr 479 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 ω) = ω)
2322oveq1d 6542 . . . . 5 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (ω +𝑜 𝑥))
2418, 23eqtr3d 2645 . . . 4 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥))
25 oveq2 6535 . . . . 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 2624 . . . 4 ((ω +𝑜 𝑥) = 𝐵 → ((𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) = 𝐵))
2824, 27syl5ibcom 233 . . 3 ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +𝑜 𝑥) = 𝐵 → (𝐴 +𝑜 𝐵) = 𝐵))
2928rexlimdva 3012 . 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 382   = wceq 1474  wcel 1976  wrex 2896  ∃!wreu 2897  Vcvv 3172  wss 3539  Oncon0 5626  (class class class)co 6527  ωcom 6934   +𝑜 coa 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-oadd 7428
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator