Theorem oaabs 8653

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 5323	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐵 ∧ 𝐵 ∈ On) → ω ∈ V)
2	1	ex 412	. . . . . . . 8 ⊢ (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3		omelon2 7872	. . . . . . . 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 711	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7		simplr 766	. . . . 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 8561	. . . 4 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
10	8, 9	sylancom 587	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
11		reurex 3379	. . 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 7865	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
14	13	ad3antrrr 727	. . . . . 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 8567	. . . . . 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 764	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20		oaabslem 8652	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
21	6, 19, 20	syl2anc 583	. . . . . . 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 7427	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (ω +o 𝑥))
24	18, 23	eqtr3d 2773	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥))
25		oveq2 7420	. . . . 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 2747	. . . 4 ⊢ ((ω +o 𝑥) = 𝐵 → ((𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥) ↔ (𝐴 +o 𝐵) = 𝐵))
28	24, 27	syl5ibcom 244	. . 3 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +o 𝑥) = 𝐵 → (𝐴 +o 𝐵) = 𝐵))
29	28	rexlimdva 3154	. 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 1540   ∈ wcel 2105  ∃wrex 3069  ∃!wreu 3373  Vcvv 3473   ⊆ wss 3948  Oncon0 6364  (class class class)co 7412  ωcom 7859   +_o coa 8469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476

This theorem is referenced by:  omabs2  42544  naddwordnexlem4  42614