Theorem oaabs 8685

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.

Step	Hyp	Ref	Expression
1		ssexg 5329	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐵 ∧ 𝐵 ∈ On) → ω ∈ V)
2	1	ex 412	. . . . . . . 8 ⊢ (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3		omelon2 7900	. . . . . . . 8 ⊢ (ω ∈ V → ω ∈ On)
4	2, 3	syl6com 37	. . . . . . 7 ⊢ (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
5	4	imp 406	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
6	5	adantll 714	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7		simplr 769	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
8	6, 7	jca 511	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9		oawordeu 8592	. . . 4 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
10	8, 9	sylancom 588	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
11		reurex 3382	. . 3 ⊢ (∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
12	10, 11	syl 17	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
13		nnon 7893	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
14	13	ad3antrrr 730	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
15	6	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16		simpr 484	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17		oaass 8598	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
18	14, 15, 16, 17	syl3anc 1370	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
19		simpll 767	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20		oaabslem 8684	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
21	6, 19, 20	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o ω) = ω)
22	21	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o ω) = ω)
23	22	oveq1d 7446	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (ω +o 𝑥))
24	18, 23	eqtr3d 2777	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥))
25		oveq2 7439	. . . . 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)
27	25, 26	eqeq12d 2751	. . . 4 ⊢ ((ω +o 𝑥) = 𝐵 → ((𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
28	24, 27	syl5ibcom 245	. . 3 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
29	28	rexlimdva 3153	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (∃𝑥 ∈ On (ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
30	12, 29	mpd 15	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∃wrex 3068  ∃!wreu 3376  Vcvv 3478   ⊆ wss 3963  Oncon0 6386  (class class class)co 7431  ωcom 7887   +_o coa 8502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  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 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509

This theorem is referenced by:  omabs2  43322  naddwordnexlem4  43391