Theorem oesuclem 7809

Description: Lemma for oesuc 7811. (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 6848	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 suc 𝐵) = (∅ ↑𝑜 suc 𝐵))
2		oesuclem.1	. . . . . . . 8 ⊢ Lim 𝑋
3		limord 5966	. . . . . . . 8 ⊢ (Lim 𝑋 → Ord 𝑋)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Ord 𝑋
5		ordelord 5929	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → Ord 𝐵)
6	4, 5	mpan 681	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → Ord 𝐵)
7		0elsuc 7232	. . . . . 6 ⊢ (Ord 𝐵 → ∅ ∈ suc 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → ∅ ∈ suc 𝐵)
9		limsuc 7246	. . . . . . 7 ⊢ (Lim 𝑋 → (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋)
11		ordelon 5931	. . . . . . . 8 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ∈ On)
12	4, 11	mpan 681	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
13		oe0m1 7805	. . . . . . 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 208	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑𝑜 suc 𝐵) = ∅))
16	8, 15	mpbid 223	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑𝑜 suc 𝐵) = ∅)
17	1, 16	sylan9eqr 2820	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑𝑜 suc 𝐵) = ∅)
18		oveq1 6848	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
19		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
20	18, 19	oveq12d 6859	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴) = ((∅ ↑𝑜 𝐵) ·𝑜 ∅))
21		ordelon 5931	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ On)
22	4, 21	mpan 681	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ On)
23		oveq2 6849	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) = (∅ ↑𝑜 ∅))
24		oe0m0 7804	. . . . . . . . . 10 ⊢ (∅ ↑𝑜 ∅) = 1𝑜
25		1on 7770	. . . . . . . . . 10 ⊢ 1𝑜 ∈ On
26	24, 25	eqeltri 2839	. . . . . . . . 9 ⊢ (∅ ↑𝑜 ∅) ∈ On
27	23, 26	syl6eqel 2851	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑𝑜 𝐵) ∈ On)
28	27	adantl 473	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐵 = ∅) → (∅ ↑𝑜 𝐵) ∈ On)
29		oe0m1 7805	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
30	22, 29	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
31	30	biimpa 468	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
32		0elon 5960	. . . . . . . . 9 ⊢ ∅ ∈ On
33	31, 32	syl6eqel 2851	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
34	33	adantll 705	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) ∈ On)
35	28, 34	oe0lem 7797	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) → (∅ ↑𝑜 𝐵) ∈ On)
36	22, 35	mpancom 679	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑𝑜 𝐵) ∈ On)
37		om0 7801	. . . . 5 ⊢ ((∅ ↑𝑜 𝐵) ∈ On → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
38	36, 37	syl 17	. . . 4 ⊢ (𝐵 ∈ 𝑋 → ((∅ ↑𝑜 𝐵) ·𝑜 ∅) = ∅)
39	20, 38	sylan9eqr 2820	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴) = ∅)
40	17, 39	eqtr4d 2801	. 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 718	. . 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 208	. . . 4 ⊢ (𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
44		oevn0 7799	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
45	43, 44	sylanl2 671	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘suc 𝐵))
46		ovex 6873	. . . . 5 ⊢ (𝐴 ↑𝑜 𝐵) ∈ V
47		oveq1 6848	. . . . . 6 ⊢ (𝑥 = (𝐴 ↑𝑜 𝐵) → (𝑥 ·𝑜 𝐴) = ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴))
48		eqid 2764	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴))
49		ovex 6873	. . . . . 6 ⊢ ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴) ∈ V
50	47, 48, 49	fvmpt 6470	. . . . 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 7799	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
53	22, 52	sylanl2 671	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
54	53	fveq2d 6378	. . . 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	syl5eqr 2812	. . 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 2808	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑𝑜 suc 𝐵) = ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴))
57	40, 56	oe0lem 7797	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑𝑜 suc 𝐵) = ((𝐴 ↑𝑜 𝐵) ·𝑜 𝐴))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1652   ∈ wcel 2155  Vcvv 3349  ∅c0 4078   ↦ cmpt 4887  Ord word 5906  Oncon0 5907  Lim wlim 5908  suc csuc 5909  ‘cfv 6067  (class class class)co 6841  reccrdg 7708  1_𝑜c1o 7756   ·_𝑜 comu 7761   ↑_𝑜 coe 7762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-omul 7768  df-oexp 7769

This theorem is referenced by:  oesuc  7811  onesuc  7814