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Theorem relexp0a 37475
 Description: Absorbtion 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 11239 . . 3 (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
2 oveq2 6613 . . . . . . . 8 (𝑥 = 1 → (𝐴𝑟𝑥) = (𝐴𝑟1))
32oveq1d 6620 . . . . . . 7 (𝑥 = 1 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟1)↑𝑟0))
43sseq1d 3616 . . . . . 6 (𝑥 = 1 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0)))
54imbi2d 330 . . . . 5 (𝑥 = 1 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))))
6 oveq2 6613 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑟𝑥) = (𝐴𝑟𝑦))
76oveq1d 6620 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑦)↑𝑟0))
87sseq1d 3616 . . . . . 6 (𝑥 = 𝑦 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)))
98imbi2d 330 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))))
10 oveq2 6613 . . . . . . . 8 (𝑥 = (𝑦 + 1) → (𝐴𝑟𝑥) = (𝐴𝑟(𝑦 + 1)))
1110oveq1d 6620 . . . . . . 7 (𝑥 = (𝑦 + 1) → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟(𝑦 + 1))↑𝑟0))
1211sseq1d 3616 . . . . . 6 (𝑥 = (𝑦 + 1) → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0)))
1312imbi2d 330 . . . . 5 (𝑥 = (𝑦 + 1) → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))))
14 oveq2 6613 . . . . . . . 8 (𝑥 = 𝑁 → (𝐴𝑟𝑥) = (𝐴𝑟𝑁))
1514oveq1d 6620 . . . . . . 7 (𝑥 = 𝑁 → ((𝐴𝑟𝑥)↑𝑟0) = ((𝐴𝑟𝑁)↑𝑟0))
1615sseq1d 3616 . . . . . 6 (𝑥 = 𝑁 → (((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0) ↔ ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
1716imbi2d 330 . . . . 5 (𝑥 = 𝑁 → ((𝐴𝑉 → ((𝐴𝑟𝑥)↑𝑟0) ⊆ (𝐴𝑟0)) ↔ (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))))
18 relexp1g 13695 . . . . . . 7 (𝐴𝑉 → (𝐴𝑟1) = 𝐴)
1918oveq1d 6620 . . . . . 6 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) = (𝐴𝑟0))
20 ssid 3608 . . . . . 6 (𝐴𝑟0) ⊆ (𝐴𝑟0)
2119, 20syl6eqss 3639 . . . . 5 (𝐴𝑉 → ((𝐴𝑟1)↑𝑟0) ⊆ (𝐴𝑟0))
22 simp2 1060 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝐴𝑉)
23 simp1 1059 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → 𝑦 ∈ ℕ)
24 relexpsucnnr 13694 . . . . . . . . . 10 ((𝐴𝑉𝑦 ∈ ℕ) → (𝐴𝑟(𝑦 + 1)) = ((𝐴𝑟𝑦) ∘ 𝐴))
2524oveq1d 6620 . . . . . . . . 9 ((𝐴𝑉𝑦 ∈ ℕ) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
2622, 23, 25syl2anc 692 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) = (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0))
27 ovex 6633 . . . . . . . . . . . . 13 (𝐴𝑟𝑦) ∈ V
28 coexg 7067 . . . . . . . . . . . . 13 (((𝐴𝑟𝑦) ∈ V ∧ 𝐴𝑉) → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
2927, 28mpan 705 . . . . . . . . . . . 12 (𝐴𝑉 → ((𝐴𝑟𝑦) ∘ 𝐴) ∈ V)
30 relexp0g 13691 . . . . . . . . . . . 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 5349 . . . . . . . . . . . . 13 dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴
33 rncoss 5350 . . . . . . . . . . . . 13 ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)
34 unss12 3768 . . . . . . . . . . . . 13 ((dom ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ dom 𝐴 ∧ ran ((𝐴𝑟𝑦) ∘ 𝐴) ⊆ ran (𝐴𝑟𝑦)) → (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦)))
3532, 33, 34mp2an 707 . . . . . . . . . . . 12 (dom ((𝐴𝑟𝑦) ∘ 𝐴) ∪ ran ((𝐴𝑟𝑦) ∘ 𝐴)) ⊆ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))
36 ssres2 5388 . . . . . . . . . . . 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, 37syl6eqss 3639 . . . . . . . . . 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 5373 . . . . . . . . . . 11 ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) = (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦)))
41 ssun1 3759 . . . . . . . . . . . . . . 15 dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴)
42 ssres2 5388 . . . . . . . . . . . . . . 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 13691 . . . . . . . . . . . . . 14 (𝐴𝑉 → (𝐴𝑟0) = ( I ↾ (dom 𝐴 ∪ ran 𝐴)))
4543, 44syl5sseqr 3638 . . . . . . . . . . . . 13 (𝐴𝑉 → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
4645adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ dom 𝐴) ⊆ (𝐴𝑟0))
47 ssun2 3760 . . . . . . . . . . . . . . 15 ran (𝐴𝑟𝑦) ⊆ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))
48 ssres2 5388 . . . . . . . . . . . . . . 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 13691 . . . . . . . . . . . . . . 15 ((𝐴𝑟𝑦) ∈ V → ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦))))
5127, 50ax-mp 5 . . . . . . . . . . . . . 14 ((𝐴𝑟𝑦)↑𝑟0) = ( I ↾ (dom (𝐴𝑟𝑦) ∪ ran (𝐴𝑟𝑦)))
5249, 51sseqtr4i 3622 . . . . . . . . . . . . 13 ( I ↾ ran (𝐴𝑟𝑦)) ⊆ ((𝐴𝑟𝑦)↑𝑟0)
53 simpr 477 . . . . . . . . . . . . 13 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0))
5452, 53syl5ss 3599 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ ran (𝐴𝑟𝑦)) ⊆ (𝐴𝑟0))
5546, 54unssd 3772 . . . . . . . . . . 11 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (( I ↾ dom 𝐴) ∪ ( I ↾ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5640, 55syl5eqss 3633 . . . . . . . . . 10 ((𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
57563adant1 1077 . . . . . . . . 9 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ( I ↾ (dom 𝐴 ∪ ran (𝐴𝑟𝑦))) ⊆ (𝐴𝑟0))
5839, 57sstrd 3598 . . . . . . . 8 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → (((𝐴𝑟𝑦) ∘ 𝐴)↑𝑟0) ⊆ (𝐴𝑟0))
5926, 58eqsstrd 3623 . . . . . . 7 ((𝑦 ∈ ℕ ∧ 𝐴𝑉 ∧ ((𝐴𝑟𝑦)↑𝑟0) ⊆ (𝐴𝑟0)) → ((𝐴𝑟(𝑦 + 1))↑𝑟0) ⊆ (𝐴𝑟0))
60593exp 1261 . . . . . 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 10983 . . . 4 (𝑁 ∈ ℕ → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
63 oveq2 6613 . . . . . . . 8 (𝑁 = 0 → (𝐴𝑟𝑁) = (𝐴𝑟0))
6463oveq1d 6620 . . . . . . 7 (𝑁 = 0 → ((𝐴𝑟𝑁)↑𝑟0) = ((𝐴𝑟0)↑𝑟0))
65 relexp0idm 37474 . . . . . . 7 (𝐴𝑉 → ((𝐴𝑟0)↑𝑟0) = (𝐴𝑟0))
6664, 65sylan9eq 2680 . . . . . 6 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0))
67 eqimss 3641 . . . . . 6 (((𝐴𝑟𝑁)↑𝑟0) = (𝐴𝑟0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6866, 67syl 17 . . . . 5 ((𝑁 = 0 ∧ 𝐴𝑉) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
6968ex 450 . . . 4 (𝑁 = 0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7062, 69jaoi 394 . . 3 ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
711, 70sylbi 207 . 2 (𝑁 ∈ ℕ0 → (𝐴𝑉 → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0)))
7271impcom 446 1 ((𝐴𝑉𝑁 ∈ ℕ0) → ((𝐴𝑟𝑁)↑𝑟0) ⊆ (𝐴𝑟0))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1992  Vcvv 3191   ∪ cun 3558   ⊆ wss 3560   I cid 4989  dom cdm 5079  ran crn 5080   ↾ cres 5081   ∘ ccom 5083  (class class class)co 6605  0cc0 9881  1c1 9882   + caddc 9884  ℕcn 10965  ℕ0cn0 11237  ↑𝑟crelexp 13689 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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958 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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-seq 12739  df-relexp 13690 This theorem is referenced by: (None)
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