Theorem oesuclem 8317

Description: Lemma for oesuc 8319. (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 7262	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o suc 𝐵) = (∅ ↑o suc 𝐵))
2		oesuclem.1	. . . . . . . 8 ⊢ Lim 𝑋
3		limord 6310	. . . . . . . 8 ⊢ (Lim 𝑋 → Ord 𝑋)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Ord 𝑋
5		ordelord 6273	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → Ord 𝐵)
6	4, 5	mpan 686	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → Ord 𝐵)
7		0elsuc 7657	. . . . . 6 ⊢ (Ord 𝐵 → ∅ ∈ suc 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → ∅ ∈ suc 𝐵)
9		limsuc 7671	. . . . . . 7 ⊢ (Lim 𝑋 → (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋)
11		ordelon 6275	. . . . . . . 8 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ∈ On)
12	4, 11	mpan 686	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
13		oe0m1 8313	. . . . . . 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 216	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
16	8, 15	mpbid 231	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o suc 𝐵) = ∅)
17	1, 16	sylan9eqr 2801	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑o suc 𝐵) = ∅)
18		oveq1 7262	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
19		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
20	18, 19	oveq12d 7273	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21		ordelon 6275	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ On)
22	4, 21	mpan 686	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ On)
23		oveq2 7263	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24		oe0m0 8312	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
25		1on 8274	. . . . . . . . . 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 8313	. . . . . . . . . . 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 6304	. . . . . . . . 9 ⊢ ∅ ∈ On
33	31, 32	eqeltrdi 2847	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
34	33	adantll 710	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
35	28, 34	oe0lem 8305	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) → (∅ ↑o 𝐵) ∈ On)
36	22, 35	mpancom 684	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o 𝐵) ∈ On)
37		om0 8309	. . . . 5 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
38	36, 37	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
39	20, 38	sylan9eqr 2801	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → ((𝐴 ↑o 𝐵) ·o 𝐴) = ∅)
40	17, 39	eqtr4d 2781	. 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 723	. . 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 216	. . . 4 ⊢ (𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
44		oevn0 8307	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
45	43, 44	sylanl2 677	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
46		ovex 7288	. . . . 5 ⊢ (𝐴 ↑o 𝐵) ∈ V
47		oveq1 7262	. . . . . 6 ⊢ (𝑥 = (𝐴 ↑o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴 ↑o 𝐵) ·o 𝐴))
48		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49		ovex 7288	. . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ·o 𝐴) ∈ V
50	47, 48, 49	fvmpt 6857	. . . . 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 8307	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
53	22, 52	sylanl2 677	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
54	53	fveq2d 6760	. . . 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 2793	. . 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 2788	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
57	40, 56	oe0lem 8305	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253   ↦ cmpt 5153  Ord word 6250  Oncon0 6251  Lim wlim 6252  suc csuc 6253  ‘cfv 6418  (class class class)co 7255  reccrdg 8211  1_oc1o 8260   ·_o comu 8265   ↑_o coe 8266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-omul 8272  df-oexp 8273

This theorem is referenced by:  oesuc  8319  onesuc  8322