Theorem oaabs 8658

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 5280	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐵 ∧ 𝐵 ∈ On) → ω ∈ V)
2	1	ex 418	. . . . . . . 8 ⊢ (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3		omelon2 7890	. . . . . . . 8 ⊢ (ω ∈ V → ω ∈ On)
4	2, 3	syl6com 38	. . . . . . 7 ⊢ (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
5	4	imp 412	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
6	5	adantll 727	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7		simplr 781	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
8	6, 7	jca 521	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9		oawordeu 8564	. . . 4 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
10	8, 9	sylancom 600	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
11		reurex 3369	. . 3 ⊢ (∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
12	10, 11	syl 18	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃𝑥 ∈ On (ω +o 𝑥) = 𝐵)
13		nnon 7883	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
14	13	ad3antrrr 743	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
15	6	adantr 486	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16		simpr 490	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17		oaass 8570	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
18	14, 15, 16, 17	syl3anc 1398	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
19		simpll 779	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20		oaabslem 8657	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
21	6, 19, 20	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o ω) = ω)
22	21	adantr 486	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o ω) = ω)
23	22	oveq1d 7435	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (ω +o 𝑥))
24	18, 23	eqtr3d 2797	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥))
25		oveq2 7428	. . . . 5 ⊢ ((ω +o 𝑥) = 𝐵 → (𝐴 +o (ω +o 𝑥)) = (𝐴 +o 𝐵))
26		id 23	. . . . 5 ⊢ ((ω +o 𝑥) = 𝐵 → (ω +o 𝑥) = 𝐵)
27	25, 26	eqeq12d 2776	. . . 4 ⊢ ((ω +o 𝑥) = 𝐵 → ((𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
28	24, 27	syl5ibcom 248	. . 3 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
29	28	rexlimdva 3163	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (∃𝑥 ∈ On (ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
30	12, 29	mpd 16	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086  ∃!wreu 3363  Vcvv 3450   ⊆ wss 3895  Oncon0 6362  (class class class)co 7420  ωcom 7877   +_o coa 8474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-oadd 8481

This theorem is used by:  omabs2  44410  naddwordnexlem4  44479