Theorem oe1m 7512

Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Assertion

Ref	Expression
oe1m	⊢ (𝐴 ∈ On → (1𝑜 ↑𝑜 𝐴) = 1𝑜)

Proof of Theorem oe1m

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . 3 ⊢ (𝑥 = ∅ → (1𝑜 ↑𝑜 𝑥) = (1𝑜 ↑𝑜 ∅))
2	1	eqeq1d 2612	. 2 ⊢ (𝑥 = ∅ → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ (1𝑜 ↑𝑜 ∅) = 1𝑜))
3		oveq2 6557	. . 3 ⊢ (𝑥 = 𝑦 → (1𝑜 ↑𝑜 𝑥) = (1𝑜 ↑𝑜 𝑦))
4	3	eqeq1d 2612	. 2 ⊢ (𝑥 = 𝑦 → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ (1𝑜 ↑𝑜 𝑦) = 1𝑜))
5		oveq2 6557	. . 3 ⊢ (𝑥 = suc 𝑦 → (1𝑜 ↑𝑜 𝑥) = (1𝑜 ↑𝑜 suc 𝑦))
6	5	eqeq1d 2612	. 2 ⊢ (𝑥 = suc 𝑦 → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ (1𝑜 ↑𝑜 suc 𝑦) = 1𝑜))
7		oveq2 6557	. . 3 ⊢ (𝑥 = 𝐴 → (1𝑜 ↑𝑜 𝑥) = (1𝑜 ↑𝑜 𝐴))
8	7	eqeq1d 2612	. 2 ⊢ (𝑥 = 𝐴 → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ (1𝑜 ↑𝑜 𝐴) = 1𝑜))
9		1on 7454	. . 3 ⊢ 1𝑜 ∈ On
10		oe0 7489	. . 3 ⊢ (1𝑜 ∈ On → (1𝑜 ↑𝑜 ∅) = 1𝑜)
11	9, 10	ax-mp 5	. 2 ⊢ (1𝑜 ↑𝑜 ∅) = 1𝑜
12		oesuc 7494	. . . . 5 ⊢ ((1𝑜 ∈ On ∧ 𝑦 ∈ On) → (1𝑜 ↑𝑜 suc 𝑦) = ((1𝑜 ↑𝑜 𝑦) ·𝑜 1𝑜))
13	9, 12	mpan 702	. . . 4 ⊢ (𝑦 ∈ On → (1𝑜 ↑𝑜 suc 𝑦) = ((1𝑜 ↑𝑜 𝑦) ·𝑜 1𝑜))
14		oveq1 6556	. . . . 5 ⊢ ((1𝑜 ↑𝑜 𝑦) = 1𝑜 → ((1𝑜 ↑𝑜 𝑦) ·𝑜 1𝑜) = (1𝑜 ·𝑜 1𝑜))
15		om1 7509	. . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ·𝑜 1𝑜) = 1𝑜)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1𝑜 ·𝑜 1𝑜) = 1𝑜
17	14, 16	syl6eq 2660	. . . 4 ⊢ ((1𝑜 ↑𝑜 𝑦) = 1𝑜 → ((1𝑜 ↑𝑜 𝑦) ·𝑜 1𝑜) = 1𝑜)
18	13, 17	sylan9eq 2664	. . 3 ⊢ ((𝑦 ∈ On ∧ (1𝑜 ↑𝑜 𝑦) = 1𝑜) → (1𝑜 ↑𝑜 suc 𝑦) = 1𝑜)
19	18	ex 449	. 2 ⊢ (𝑦 ∈ On → ((1𝑜 ↑𝑜 𝑦) = 1𝑜 → (1𝑜 ↑𝑜 suc 𝑦) = 1𝑜))
20		iuneq2 4473	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = 1𝑜 → ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 1𝑜)
21		vex 3176	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 7471	. . . . . . . 8 ⊢ ∅ ∈ 1𝑜
23		oelim 7501	. . . . . . . 8 ⊢ (((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1𝑜) → (1𝑜 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦))
24	22, 23	mpan2 703	. . . . . . 7 ⊢ ((1𝑜 ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1𝑜 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦))
25	9, 24	mpan 702	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1𝑜 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦))
26	21, 25	mpan 702	. . . . 5 ⊢ (Lim 𝑥 → (1𝑜 ↑𝑜 𝑥) = ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦))
27	26	eqeq1d 2612	. . . 4 ⊢ (Lim 𝑥 → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = 1𝑜))
28		0ellim 5704	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 3880	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4465	. . . . . 6 ⊢ (𝑥 ≠ ∅ → ∪ 𝑦 ∈ 𝑥 1𝑜 = 1𝑜)
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (Lim 𝑥 → ∪ 𝑦 ∈ 𝑥 1𝑜 = 1𝑜)
32	31	eqeq2d 2620	. . . 4 ⊢ (Lim 𝑥 → (∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 1𝑜 ↔ ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = 1𝑜))
33	27, 32	bitr4d 270	. . 3 ⊢ (Lim 𝑥 → ((1𝑜 ↑𝑜 𝑥) = 1𝑜 ↔ ∪ 𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = ∪ 𝑦 ∈ 𝑥 1𝑜))
34	20, 33	syl5ibr 235	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1𝑜 ↑𝑜 𝑦) = 1𝑜 → (1𝑜 ↑𝑜 𝑥) = 1𝑜))
35	2, 4, 6, 8, 11, 19, 34	tfinds 6951	1 ⊢ (𝐴 ∈ On → (1𝑜 ↑𝑜 𝐴) = 1𝑜)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  ∪ ciun 4455  Oncon0 5640  Lim wlim 5641  suc csuc 5642  (class class class)co 6549  1_𝑜c1o 7440   ·_𝑜 comu 7445   ↑_𝑜 coe 7446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-oexp 7453

This theorem is referenced by:  oewordi  7558  oeoe  7566  cantnflem2  8470