Theorem oesuclem 8495

Description: Lemma for oesuc 8497. (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 7404	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ↑o suc 𝐵) = (∅ ↑o suc 𝐵))
2		oesuclem.1	. . . . . . . 8 ⊢ Lim 𝑋
3		limord 6408	. . . . . . . 8 ⊢ (Lim 𝑋 → Ord 𝑋)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ Ord 𝑋
5		ordelord 6369	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → Ord 𝐵)
6	4, 5	mpan 700	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → Ord 𝐵)
7		0elsuc 7816	. . . . . 6 ⊢ (Ord 𝐵 → ∅ ∈ suc 𝐵)
8	6, 7	syl 17	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → ∅ ∈ suc 𝐵)
9		limsuc 7830	. . . . . . 7 ⊢ (Lim 𝑋 → (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 ↔ suc 𝐵 ∈ 𝑋)
11		ordelon 6371	. . . . . . . 8 ⊢ ((Ord 𝑋 ∧ suc 𝐵 ∈ 𝑋) → suc 𝐵 ∈ On)
12	4, 11	mpan 700	. . . . . . 7 ⊢ (suc 𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
13		oe0m1 8491	. . . . . . 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 219	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ suc 𝐵 ↔ (∅ ↑o suc 𝐵) = ∅))
16	8, 15	mpbid 234	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o suc 𝐵) = ∅)
17	1, 16	sylan9eqr 2820	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐴 = ∅) → (𝐴 ↑o suc 𝐵) = ∅)
18		oveq1 7404	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
19		id 22	. . . . 5 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
20	18, 19	oveq12d 7415	. . . 4 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ·o 𝐴) = ((∅ ↑o 𝐵) ·o ∅))
21		ordelon 6371	. . . . . . 7 ⊢ ((Ord 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ On)
22	4, 21	mpan 700	. . . . . 6 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ On)
23		oveq2 7405	. . . . . . . . 9 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) = (∅ ↑o ∅))
24		oe0m0 8490	. . . . . . . . . 10 ⊢ (∅ ↑o ∅) = 1o
25		1on 8451	. . . . . . . . . 10 ⊢ 1o ∈ On
26	24, 25	eqeltri 2859	. . . . . . . . 9 ⊢ (∅ ↑o ∅) ∈ On
27	23, 26	eqeltrdi 2871	. . . . . . . 8 ⊢ (𝐵 = ∅ → (∅ ↑o 𝐵) ∈ On)
28	27	adantl 485	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐵 = ∅) → (∅ ↑o 𝐵) ∈ On)
29		oe0m1 8491	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
30	22, 29	syl 17	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → (∅ ∈ 𝐵 ↔ (∅ ↑o 𝐵) = ∅))
31	30	biimpa 480	. . . . . . . . 9 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) = ∅)
32		0elon 6402	. . . . . . . . 9 ⊢ ∅ ∈ On
33	31, 32	eqeltrdi 2871	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑋 ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
34	33	adantll 724	. . . . . . 7 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐵) → (∅ ↑o 𝐵) ∈ On)
35	28, 34	oe0lem 8483	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ 𝑋) → (∅ ↑o 𝐵) ∈ On)
36	22, 35	mpancom 698	. . . . 5 ⊢ (𝐵 ∈ 𝑋 → (∅ ↑o 𝐵) ∈ On)
37		om0 8487	. . . . 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 737	. . 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 219	. . . 4 ⊢ (𝐵 ∈ 𝑋 → suc 𝐵 ∈ On)
44		oevn0 8485	. . . 4 ⊢ (((𝐴 ∈ On ∧ suc 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
45	43, 44	sylanl2 691	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o suc 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘suc 𝐵))
46		ovex 7430	. . . . 5 ⊢ (𝐴 ↑o 𝐵) ∈ V
47		oveq1 7404	. . . . . 6 ⊢ (𝑥 = (𝐴 ↑o 𝐵) → (𝑥 ·o 𝐴) = ((𝐴 ↑o 𝐵) ·o 𝐴))
48		eqid 2763	. . . . . 6 ⊢ (𝑥 ∈ V ↦ (𝑥 ·o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 ·o 𝐴))
49		ovex 7430	. . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ·o 𝐴) ∈ V
50	47, 48, 49	fvmpt 6976	. . . . 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 8485	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
53	22, 52	sylanl2 691	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) ∧ ∅ ∈ 𝐴) → (𝐴 ↑o 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·o 𝐴)), 1o)‘𝐵))
54	53	fveq2d 6872	. . . 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 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 8483	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝑋) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143  Vcvv 3455  ∅c0 4286   ↦ cmpt 5182  Ord word 6346  Oncon0 6347  Lim wlim 6348  suc csuc 6349  ‘cfv 6522  (class class class)co 7397  reccrdg 8381  1_oc1o 8431   ·_o comu 8436   ↑_o coe 8437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-omul 8443  df-oexp 8444

This theorem is referenced by:  oesuc  8497  onesuc  8500