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Theorem relexp0a 42769
Description: Absorption law for zeroth power of a relation. (Contributed by RP, 17-Jun-2020.)
Assertion
Ref Expression
relexp0a ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))

Proof of Theorem relexp0a
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elnn0 12478 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 7419 . . . . . . . 8 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
32oveq1d 7426 . . . . . . 7 (𝑥 = 1 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟1)↑𝑟0))
43sseq1d 4012 . . . . . 6 (𝑥 = 1 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0)))
54imbi2d 339 . . . . 5 (𝑥 = 1 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))))
6 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
76oveq1d 7426 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑦)↑𝑟0))
87sseq1d 4012 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)))
98imbi2d 339 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))))
10 oveq2 7419 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
1110oveq1d 7426 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟(𝑦 + 1))↑𝑟0))
1211sseq1d 4012 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0)))
1312imbi2d 339 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
14 oveq2 7419 . . . . . . . 8 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
1514oveq1d 7426 . . . . . . 7 (𝑥 = 𝑁 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑁)↑𝑟0))
1615sseq1d 4012 . . . . . 6 (𝑥 = 𝑁 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
1716imbi2d 339 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))))
18 relexp1g 14977 . . . . . . 7 (𝐴𝑉 → (𝐴𝑟1) = 𝐴)
1918oveq1d 7426 . . . . . 6 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) = (𝐴𝑟0))
20 ssid 4003 . . . . . 6 (𝐴𝑟0) ⊆ (𝐴𝑟0)
2119, 20eqsstrdi 4035 . . . . 5 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))
22 simp2 1135 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝐴𝑉)
23 simp1 1134 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝑦 ∈ ℕ)
24 relexpsucnnr 14976 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
2524oveq1d 7426 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ ℕ) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
2622, 23, 25syl2anc 582 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
27 ovex 7444 . . . . . . . . . . . . 13 (𝐴𝑟𝑦) ∈ V
28 coexg 7922 . . . . . . . . . . . . 13 (((𝐴𝑟𝑦) ∈ V ∧ 𝐴𝑉) → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
2927, 28mpan 686 . . . . . . . . . . . 12 (𝐴𝑉 → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
30 relexp0g 14973 . . . . . . . . . . . 12 (((𝐴𝑟𝑦) ∘ 𝐴) ∈ V → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) = ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))))
3129, 30syl 17 . . . . . . . . . . 11 (𝐴𝑉 → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) = ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))))
32 dmcoss 5969 . . . . . . . . . . . . 13 dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33 rncoss 5970 . . . . . . . . . . . . 13 ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)
34 unss12 4181 . . . . . . . . . . . . 13 ((dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴 ∧ ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)) → (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)))
3532, 33, 34mp2an 688 . . . . . . . . . . . 12 (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))
36 ssres2 6008 . . . . . . . . . . . 12 ((dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)) → ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
3735, 36ax-mp 5 . . . . . . . . . . 11 ( I ↾ (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴))) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)))
3831, 37eqsstrdi 4035 . . . . . . . . . 10 (𝐴𝑉 → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
3922, 38syl 17 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))))
40 resundi 5994 . . . . . . . . . . 11 ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦)))
41 ssun1 4171 . . . . . . . . . . . . . . 15 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42 ssres2 6008 . . . . . . . . . . . . . . 15 (dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) → ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4341, 42ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ dom 𝐴) ⊆ ( I ↾ (dom 𝐴 ∪ ran 𝐴))
44 relexp0g 14973 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4543, 44sseqtrrid 4034 . . . . . . . . . . . . 13 (𝐴𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
4645adantr 479 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
47 ssun2 4172 . . . . . . . . . . . . . . 15 ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))
48 ssres2 6008 . . . . . . . . . . . . . . 15 (ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
4947, 48ax-mp 5 . . . . . . . . . . . . . 14 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
50 relexp0g 14973 . . . . . . . . . . . . . . 15 ((𝐴𝑟𝑦) ∈ V → ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
5127, 50ax-mp 5 . . . . . . . . . . . . . 14 ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
5249, 51sseqtrri 4018 . . . . . . . . . . . . 13 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ((𝐴𝑟𝑦)↑𝑟0)
53 simpr 483 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))
5452, 53sstrid 3992 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ (𝐴𝑟0))
5546, 54unssd 4185 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5640, 55eqsstrid 4029 . . . . . . . . . 10 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
57563adant1 1128 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5839, 57sstrd 3991 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ (𝐴𝑟0))
5926, 58eqsstrd 4019 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))
60593exp 1117 . . . . . 6 (𝑦 ∈ ℕ → (𝐴𝑉 → (((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
6160a2d 29 . . . . 5 (𝑦 ∈ ℕ → ((𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
625, 9, 13, 17, 21, 61nnind 12234 . . . 4 (𝑁 ∈ ℕ → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
63 oveq2 7419 . . . . . . . 8 (𝑁 = 0 → (𝐴𝑟𝑁) = (𝐴𝑟0))
6463oveq1d 7426 . . . . . . 7 (𝑁 = 0 → ((𝐴𝑟𝑁)↑𝑟0) = ((𝐴𝑟0)↑𝑟0))
65 relexp0idm 42768 . . . . . . 7 (𝐴𝑉 → ((𝐴𝑟0)↑𝑟0) = (𝐴𝑟0))
6664, 65sylan9eq 2790 . . . . . 6 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0))
67 eqimss 4039 . . . . . 6 (((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6866, 67syl 17 . . . . 5 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6968ex 411 . . . 4 (𝑁 = 0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7062, 69jaoi 853 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
711, 70sylbi 216 . 2 (𝑁 ∈ ℕ0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7271impcom 406 1 ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 843  w3a 1085   = wceq 1539  wcel 2104  Vcvv 3472  cun 3945  wss 3947   I cid 5572  dom cdm 5675  ran crn 5676  cres 5677  ccom 5679  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115  cn 12216  0cn0 12476  𝑟crelexp 14970
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-relexp 14971
This theorem is referenced by: (None)
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