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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 12233 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | simp3 1137 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
3 | simp1 1135 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) | |
4 | relexpsucnnl 14737 | . . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) | |
5 | 2, 3, 4 | syl2anc 584 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
6 | 5 | 3expib 1121 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
7 | simp2 1136 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → Rel 𝑅) | |
8 | relcoi1 6179 | . . . . . . . 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 1135 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) | |
11 | 10 | oveq2d 7285 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
12 | simp3 1137 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
13 | relexp0 14730 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | |
14 | 12, 7, 13 | syl2anc 584 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
15 | 11, 14 | eqtrd 2780 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ ∪ ∪ 𝑅)) |
16 | 15 | coeq2d 5769 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅 ∘ (𝑅↑𝑟𝑁)) = (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅))) |
17 | 10 | oveq1d 7284 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = (0 + 1)) |
18 | 0p1e1 12093 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
19 | 17, 18 | eqtrdi 2796 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = 1) |
20 | 19 | oveq2d 7285 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅↑𝑟1)) |
21 | relexp1g 14733 | . . . . . . . . 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 2780 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = 𝑅) |
24 | 9, 16, 23 | 3eqtr4rd 2791 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
25 | 24 | 3expib 1121 | . . . . 5 ⊢ (𝑁 = 0 → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁)))) |
26 | 6, 25 | jaoi 854 | . . . 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 1115 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑁))) |
29 | 28 | 3com13 1123 | 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 396 ∨ wo 844 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 ∪ cuni 4845 I cid 5488 ↾ cres 5591 ∘ ccom 5593 Rel wrel 5594 (class class class)co 7269 0cc0 10870 1c1 10871 + caddc 10873 ℕcn 11971 ℕ0cn0 12231 ↑𝑟crelexp 14726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12580 df-seq 13718 df-relexp 14727 |
This theorem is referenced by: relexpsucld 14741 |
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