Theorem oaabs 8622

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 5284	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐵 ∧ 𝐵 ∈ On) → ω ∈ V)
2	1	ex 417	. . . . . . . 8 ⊢ (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3		omelon2 7863	. . . . . . . 8 ⊢ (ω ∈ V → ω ∈ On)
4	2, 3	syl6com 38	. . . . . . 7 ⊢ (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
5	4	imp 411	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
6	5	adantll 726	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7		simplr 780	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
8	6, 7	jca 520	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9		oawordeu 8528	. . . 4 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
10	8, 9	sylancom 599	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +o 𝑥) = 𝐵)
11		reurex 3374	. . 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 7856	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
14	13	ad3antrrr 742	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
15	6	adantr 485	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16		simpr 489	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17		oaass 8534	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
18	14, 15, 16, 17	syl3anc 1394	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (𝐴 +o (ω +o 𝑥)))
19		simpll 778	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20		oaabslem 8621	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω)
21	6, 19, 20	syl2anc 595	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o ω) = ω)
22	21	adantr 485	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o ω) = ω)
23	22	oveq1d 7415	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +o ω) +o 𝑥) = (ω +o 𝑥))
24	18, 23	eqtr3d 2802	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +o (ω +o 𝑥)) = (ω +o 𝑥))
25		oveq2 7408	. . . . 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 2781	. . . 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 3166	. 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)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  ∃!wreu 3368  Vcvv 3457   ⊆ wss 3907  Oncon0 6350  (class class class)co 7400  ωcom 7850   +_o coa 8438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-oadd 8445

This theorem is referenced by:  omabs2  43921  naddwordnexlem4  43990