Theorem oesuclem 8443

Description: Lemma for oesuc 8445. (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 7356	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o suc 𝐵) = (∅ ↑o suc 𝐵))
2		oesuclem.1	. . . . . . . 8 ⊢ Lim 𝑋
3		limord 6368	. . . . . . . 8 ⊢ (Lim 𝑋 → Ord 𝑋)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Ord 𝑋
5		ordelord 6329	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → Ord 𝐵)
6	4, 5	mpan 690	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → Ord 𝐵)
7		0elsuc 7768	. . . . . 6 ⊢ (Ord 𝐵 → ∅ ∈ suc 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → ∅ ∈ suc 𝐵)
9		limsuc 7782	. . . . . . 7 ⊢ (Lim 𝑋 → (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋)
11		ordelon 6331	. . . . . . . 8 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ∈ On)
12	4, 11	mpan 690	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
13		oe0m1 8439	. . . . . . 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 2786	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑o suc 𝐵) = ∅)
18		oveq1 7356	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
19		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
20	18, 19	oveq12d 7367	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21		ordelon 6331	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ On)
22	4, 21	mpan 690	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ On)
23		oveq2 7357	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24		oe0m0 8438	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
25		1on 8400	. . . . . . . . . 10 ⊢ 1o ∈ On
26	24, 25	eqeltri 2824	. . . . . . . . 9 ⊢ (∅ ↑o ∅) ∈ On
27	23, 26	eqeltrdi 2836	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
28	27	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29		oe0m1 8439	. . . . . . . . . . 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 6362	. . . . . . . . 9 ⊢ ∅ ∈ On
33	31, 32	eqeltrdi 2836	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
34	33	adantll 714	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
35	28, 34	oe0lem 8431	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) → (∅ ↑o 𝐵) ∈ On)
36	22, 35	mpancom 688	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o 𝐵) ∈ On)
37		om0 8435	. . . . 5 ⊢ ((∅ ↑o 𝐵) ∈ On → ((∅ ↑o 𝐵) ·o ∅) = ∅)
38	36, 37	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ((∅ ↑o 𝐵) ·o ∅) = ∅)
39	20, 38	sylan9eqr 2786	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → ((𝐴 ↑o 𝐵) ·o 𝐴) = ∅)
40	17, 39	eqtr4d 2767	. 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 8433	. . . 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 7382	. . . . 5 ⊢ (𝐴 ↑o 𝐵) ∈ V
47		oveq1 7356	. . . . . 6 ⊢ (𝑥 = (𝐴 ↑o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴 ↑o 𝐵) ·o 𝐴))
48		eqid 2729	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49		ovex 7382	. . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ·o 𝐴) ∈ V
50	47, 48, 49	fvmpt 6930	. . . . 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 8433	. . . . . 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 6826	. . . 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 2778	. . 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 2774	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))
57	40, 56	oe0lem 8431	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173  Ord word 6306  Oncon0 6307  Lim wlim 6308  suc csuc 6309  ‘cfv 6482  (class class class)co 7349  reccrdg 8331  1_oc1o 8381   ·_o comu 8386   ↑_o coe 8387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-omul 8393  df-oexp 8394

This theorem is referenced by:  oesuc  8445  onesuc  8448