Theorem oaabs 7724

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 4804	. . . . . . . . 9 ⊢ ((ω ⊆ 𝐵 ∧ 𝐵 ∈ On) → ω ∈ V)
2	1	ex 450	. . . . . . . 8 ⊢ (ω ⊆ 𝐵 → (𝐵 ∈ On → ω ∈ V))
3		omelon2 7077	. . . . . . . 8 ⊢ (ω ∈ V → ω ∈ On)
4	2, 3	syl6com 37	. . . . . . 7 ⊢ (𝐵 ∈ On → (ω ⊆ 𝐵 → ω ∈ On))
5	4	imp 445	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ω ⊆ 𝐵) → ω ∈ On)
6	5	adantll 750	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ω ∈ On)
7		simplr 792	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐵 ∈ On)
8	6, 7	jca 554	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (ω ∈ On ∧ 𝐵 ∈ On))
9		oawordeu 7635	. . . 4 ⊢ (((ω ∈ On ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
10	8, 9	sylancom 701	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → ∃!𝑥 ∈ On (ω +𝑜 𝑥) = 𝐵)
11		reurex 3160	. . 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 7071	. . . . . . 7 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On)
14	13	ad3antrrr 766	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
15	6	adantr 481	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ω ∈ On)
16		simpr 477	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
17		oaass 7641	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ ω ∈ On ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
18	14, 15, 16, 17	syl3anc 1326	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (𝐴 +𝑜 (ω +𝑜 𝑥)))
19		simpll 790	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → 𝐴 ∈ ω)
20		oaabslem 7723	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω)
21	6, 19, 20	syl2anc 693	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +𝑜 ω) = ω)
22	21	adantr 481	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 ω) = ω)
23	22	oveq1d 6665	. . . . 5 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((𝐴 +𝑜 ω) +𝑜 𝑥) = (ω +𝑜 𝑥))
24	18, 23	eqtr3d 2658	. . . 4 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → (𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥))
25		oveq2 6658	. . . . 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 2637	. . . 4 ⊢ ((ω +𝑜 𝑥) = 𝐵 → ((𝐴 +𝑜 (ω +𝑜 𝑥)) = (ω +𝑜 𝑥) ↔ (𝐴 +𝑜 𝐵) = 𝐵))
28	24, 27	syl5ibcom 235	. . 3 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) ∧ 𝑥 ∈ On) → ((ω +𝑜 𝑥) = 𝐵 → (𝐴 +𝑜 𝐵) = 𝐵))
29	28	rexlimdva 3031	. 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 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ∃!wreu 2914  Vcvv 3200   ⊆ wss 3574  Oncon0 5723  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by: (None)