Theorem oesuclem 8562

Description: Lemma for oesuc 8564. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
oesuclem.1	⊢ Lim 𝑋
oesuclem.2	⊢ (𝐵 ∈ 𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))

Assertion

Ref	Expression
oesuclem	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑋(𝑥)

Proof of Theorem oesuclem

Step	Hyp	Ref	Expression
1		oveq1 7438	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o suc 𝐵) = (∅ ↑o suc 𝐵))
2		oesuclem.1	. . . . . . . 8 ⊢ Lim 𝑋
3		limord 6446	. . . . . . . 8 ⊢ (Lim 𝑋 → Ord 𝑋)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Ord 𝑋
5		ordelord 6408	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → Ord 𝐵)
6	4, 5	mpan 690	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → Ord 𝐵)
7		0elsuc 7855	. . . . . 6 ⊢ (Ord 𝐵 → ∅ ∈ suc 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → ∅ ∈ suc 𝐵)
9		limsuc 7870	. . . . . . 7 ⊢ (Lim 𝑋 → (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋)
11		ordelon 6410	. . . . . . . 8 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ∈ On)
12	4, 11	mpan 690	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
13		oe0m1 8558	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
14	12, 13	syl 17	. . . . . 6 ⊢ (suc 𝐵 ∈ 𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
15	10, 14	sylbi 217	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
16	8, 15	mpbid 232	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o suc 𝐵) = ∅)
17	1, 16	sylan9eqr 2797	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑o suc 𝐵) = ∅)
18		oveq1 7438	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
19		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
20	18, 19	oveq12d 7449	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21		ordelon 6410	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ On)
22	4, 21	mpan 690	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ On)
23		oveq2 7439	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24		oe0m0 8557	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
25		1on 8517	. . . . . . . . . 10 ⊢ 1o ∈ On
26	24, 25	eqeltri 2835	. . . . . . . . 9 ⊢ (∅ ↑o ∅) ∈ On
27	23, 26	eqeltrdi 2847	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29		oe0m1 8558	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
30	22, 29	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
31	30	biimpa 476	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
32		0elon 6440	. . . . . . . . 9 ⊢ ∅ ∈ On
33	31, 32	eqeltrdi 2847	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
34	33	adantll 714	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
35	28, 34	oe0lem 8550	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) → (∅ ↑o 𝐵) ∈ On)
36	22, 35	mpancom 688	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o 𝐵) ∈ On)
37		om0 8554	. . . . 5 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
38	36, 37	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
39	20, 38	sylan9eqr 2797	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → ((𝐴 ↑o 𝐵) ·o 𝐴) = ∅)
40	17, 39	eqtr4d 2778	. 2 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
41		oesuclem.2	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
42	41	ad2antlr 727	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
43	10, 12	sylbi 217	. . . 4 ⊢ (𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
44		oevn0 8552	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
45	43, 44	sylanl2 681	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
46		ovex 7464	. . . . 5 ⊢ (𝐴 ↑o 𝐵) ∈ V
47		oveq1 7438	. . . . . 6 ⊢ (𝑥 = (𝐴 ↑o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴 ↑o 𝐵) ·o 𝐴))
48		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49		ovex 7464	. . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ·o 𝐴) ∈ V
50	47, 48, 49	fvmpt 7016	. . . . 5 ⊢ ((𝐴 ↑o 𝐵) ∈ V → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(𝐴 ↑o 𝐵)) = ((𝐴 ↑o 𝐵) ·o 𝐴))
51	46, 50	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(𝐴 ↑o 𝐵)) = ((𝐴 ↑o 𝐵) ·o 𝐴)
52		oevn0 8552	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
53	22, 52	sylanl2 681	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
54	53	fveq2d 6911	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(𝐴 ↑o 𝐵)) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
55	51, 54	eqtr3id 2789	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → ((𝐴 ↑o 𝐵) ·o 𝐴) = ((𝑥 ∈ V ↦ (𝑥 ·o 𝐴))‘(rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵)))
56	42, 45, 55	3eqtr4d 2785	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
57	40, 56	oe0lem 8550	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339   ↦ cmpt 5231  Ord word 6385  Oncon0 6386  Lim wlim 6387  suc csuc 6388  ‘cfv 6563  (class class class)co 7431  reccrdg 8448  1_oc1o 8498   ·_o comu 8503   ↑_o coe 8504

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-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-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-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-1o 8505  df-omul 8510  df-oexp 8511

This theorem is referenced by:  oesuc  8564  onesuc  8567