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Theorem oev2 7548
Description: Alternate value of ordinal exponentiation. Compare oev 7539. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oev2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem oev2
StepHypRef Expression
1 oveq12 6613 . . . . . 6 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = (∅ ↑𝑜 ∅))
2 oe0m0 7545 . . . . . 6 (∅ ↑𝑜 ∅) = 1𝑜
31, 2syl6eq 2671 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = 1𝑜)
4 fveq2 6148 . . . . . . . 8 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅))
5 1on 7512 . . . . . . . . . 10 1𝑜 ∈ On
65elexi 3199 . . . . . . . . 9 1𝑜 ∈ V
76rdg0 7462 . . . . . . . 8 (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘∅) = 1𝑜
84, 7syl6eq 2671 . . . . . . 7 (𝐵 = ∅ → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = 1𝑜)
9 inteq 4443 . . . . . . . 8 (𝐵 = ∅ → 𝐵 = ∅)
10 int0 4455 . . . . . . . 8 ∅ = V
119, 10syl6eq 2671 . . . . . . 7 (𝐵 = ∅ → 𝐵 = V)
128, 11ineq12d 3793 . . . . . 6 (𝐵 = ∅ → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = (1𝑜 ∩ V))
13 inv1 3942 . . . . . . 7 (1𝑜 ∩ V) = 1𝑜
1413a1i 11 . . . . . 6 (𝐴 = ∅ → (1𝑜 ∩ V) = 1𝑜)
1512, 14sylan9eqr 2677 . . . . 5 ((𝐴 = ∅ ∧ 𝐵 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = 1𝑜)
163, 15eqtr4d 2658 . . . 4 ((𝐴 = ∅ ∧ 𝐵 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
17 oveq1 6611 . . . . . . 7 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
18 oe0m1 7546 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ (∅ ↑𝑜 𝐵) = ∅))
1918biimpa 501 . . . . . . 7 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → (∅ ↑𝑜 𝐵) = ∅)
2017, 19sylan9eqr 2677 . . . . . 6 (((𝐵 ∈ On ∧ ∅ ∈ 𝐵) ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ∅)
2120an32s 845 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴𝑜 𝐵) = ∅)
22 int0el 4473 . . . . . . . 8 (∅ ∈ 𝐵 𝐵 = ∅)
2322ineq2d 3792 . . . . . . 7 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅))
24 in0 3940 . . . . . . 7 ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ∅) = ∅
2523, 24syl6eq 2671 . . . . . 6 (∅ ∈ 𝐵 → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ∅)
2625adantl 482 . . . . 5 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵) = ∅)
2721, 26eqtr4d 2658 . . . 4 (((𝐵 ∈ On ∧ 𝐴 = ∅) ∧ ∅ ∈ 𝐵) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
2816, 27oe0lem 7538 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
29 inteq 4443 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = ∅)
3029, 10syl6eq 2671 . . . . . . . . 9 (𝐴 = ∅ → 𝐴 = V)
3130difeq2d 3706 . . . . . . . 8 (𝐴 = ∅ → (V ∖ 𝐴) = (V ∖ V))
32 difid 3922 . . . . . . . 8 (V ∖ V) = ∅
3331, 32syl6eq 2671 . . . . . . 7 (𝐴 = ∅ → (V ∖ 𝐴) = ∅)
3433uneq2d 3745 . . . . . 6 (𝐴 = ∅ → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ ∅))
35 uncom 3735 . . . . . 6 ( 𝐵 ∪ (V ∖ 𝐴)) = ((V ∖ 𝐴) ∪ 𝐵)
36 un0 3939 . . . . . 6 ( 𝐵 ∪ ∅) = 𝐵
3734, 35, 363eqtr3g 2678 . . . . 5 (𝐴 = ∅ → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3837adantl 482 . . . 4 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((V ∖ 𝐴) ∪ 𝐵) = 𝐵)
3938ineq2d 3792 . . 3 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ 𝐵))
4028, 39eqtr4d 2658 . 2 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
41 oevn0 7540 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵))
42 int0el 4473 . . . . . . . . . 10 (∅ ∈ 𝐴 𝐴 = ∅)
4342difeq2d 3706 . . . . . . . . 9 (∅ ∈ 𝐴 → (V ∖ 𝐴) = (V ∖ ∅))
44 dif0 3924 . . . . . . . . 9 (V ∖ ∅) = V
4543, 44syl6eq 2671 . . . . . . . 8 (∅ ∈ 𝐴 → (V ∖ 𝐴) = V)
4645uneq2d 3745 . . . . . . 7 (∅ ∈ 𝐴 → ( 𝐵 ∪ (V ∖ 𝐴)) = ( 𝐵 ∪ V))
47 unv 3943 . . . . . . 7 ( 𝐵 ∪ V) = V
4846, 35, 473eqtr3g 2678 . . . . . 6 (∅ ∈ 𝐴 → ((V ∖ 𝐴) ∪ 𝐵) = V)
4948adantl 482 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((V ∖ 𝐴) ∪ 𝐵) = V)
5049ineq2d 3792 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ V))
51 inv1 3942 . . . 4 ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ V) = (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵)
5250, 51syl6req 2672 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5341, 52eqtrd 2655 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
5440, 53oe0lem 7538 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) = ((rec((𝑥 ∈ V ↦ (𝑥 ·𝑜 𝐴)), 1𝑜)‘𝐵) ∩ ((V ∖ 𝐴) ∪ 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  cin 3554  c0 3891   cint 4440  cmpt 4673  Oncon0 5682  cfv 5847  (class class class)co 6604  reccrdg 7450  1𝑜c1o 7498   ·𝑜 comu 7503  𝑜 coe 7504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oexp 7511
This theorem is referenced by: (None)
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