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Mirrors > Home > MPE Home > Th. List > relexpsucl | Structured version Visualization version GIF version |
Description: A reduction for relation exponentiation to the left. (Contributed by RP, 23-May-2020.) |
Ref | Expression |
---|---|
relexpsucl | ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 12507 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | simp3 1135 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
3 | simp1 1133 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) | |
4 | relexpsucnnl 15013 | . . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | |
5 | 2, 3, 4 | syl2anc 582 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
6 | 5 | 3expib 1119 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
7 | simp2 1134 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → Rel 𝑅) | |
8 | relcoi1 6284 | . . . . . . . 8 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) |
10 | simp1 1133 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) | |
11 | 10 | oveq2d 7435 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
12 | simp3 1135 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
13 | relexp0 15006 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | |
14 | 12, 7, 13 | syl2anc 582 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
15 | 11, 14 | eqtrd 2765 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ ∪ ∪ 𝑅)) |
16 | 15 | coeq2d 5865 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ (𝑅↑𝑟𝑁)) = (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅))) |
17 | 10 | oveq1d 7434 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = (0 + 1)) |
18 | 0p1e1 12367 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
19 | 17, 18 | eqtrdi 2781 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = 1) |
20 | 19 | oveq2d 7435 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅↑𝑟1)) |
21 | relexp1g 15009 | . . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | |
22 | 12, 21 | syl 17 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
23 | 20, 22 | eqtrd 2765 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = 𝑅) |
24 | 9, 16, 23 | 3eqtr4rd 2776 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
25 | 24 | 3expib 1119 | . . . . 5 ⊢ (𝑁 = 0 → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
26 | 6, 25 | jaoi 855 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
27 | 1, 26 | sylbi 216 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
28 | 27 | 3impib 1113 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
29 | 28 | 3com13 1121 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∪ cuni 4909 I cid 5575 ↾ cres 5680 ∘ ccom 5682 Rel wrel 5683 (class class class)co 7419 0cc0 11140 1c1 11141 + caddc 11143 ℕcn 12245 ℕ0cn0 12505 ↑𝑟crelexp 15002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-seq 14003 df-relexp 15003 |
This theorem is referenced by: relexpsucld 15017 |
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