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Mirrors > Home > MPE Home > Th. List > relexpsucr | Structured version Visualization version GIF version |
Description: A reduction for relation exponentiation to the right. (Contributed by RP, 23-May-2020.) |
Ref | Expression |
---|---|
relexpsucr | ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 11621 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
2 | simp3 1174 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
3 | simp1 1172 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ) | |
4 | relexpsucnnr 14143 | . . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | |
5 | 2, 3, 4 | syl2anc 581 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
6 | 5 | 3expib 1158 | . . . . 5 ⊢ (𝑁 ∈ ℕ → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) |
7 | simp2 1173 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → Rel 𝑅) | |
8 | relcoi2 5905 | . . . . . . . . 9 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) | |
9 | 8 | eqcomd 2832 | . . . . . . . 8 ⊢ (Rel 𝑅 → 𝑅 = (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅)) |
10 | 7, 9 | syl 17 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 = (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅)) |
11 | simp1 1172 | . . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0) | |
12 | 11 | oveq1d 6921 | . . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = (0 + 1)) |
13 | 0p1e1 11481 | . . . . . . . . . 10 ⊢ (0 + 1) = 1 | |
14 | 12, 13 | syl6eq 2878 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑁 + 1) = 1) |
15 | 14 | oveq2d 6922 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = (𝑅↑𝑟1)) |
16 | simp3 1174 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉) | |
17 | relexp1g 14144 | . . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟1) = 𝑅) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟1) = 𝑅) |
19 | 15, 18 | eqtrd 2862 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = 𝑅) |
20 | 11 | oveq2d 6922 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = (𝑅↑𝑟0)) |
21 | relexp0 14141 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | |
22 | 16, 7, 21 | syl2anc 581 | . . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
23 | 20, 22 | eqtrd 2862 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟𝑁) = ( I ↾ ∪ ∪ 𝑅)) |
24 | 23 | coeq1d 5517 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → ((𝑅↑𝑟𝑁) ∘ 𝑅) = (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅)) |
25 | 10, 19, 24 | 3eqtr4d 2872 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
26 | 25 | 3expib 1158 | . . . . 5 ⊢ (𝑁 = 0 → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) |
27 | 6, 26 | jaoi 890 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) |
28 | 1, 27 | sylbi 209 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) |
29 | 28 | 3impib 1150 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ Rel 𝑅 ∧ 𝑅 ∈ 𝑉) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
30 | 29 | 3com13 1160 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 880 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∪ cuni 4659 I cid 5250 ↾ cres 5345 ∘ ccom 5347 Rel wrel 5348 (class class class)co 6906 0cc0 10253 1c1 10254 + caddc 10256 ℕcn 11351 ℕ0cn0 11619 ↑𝑟crelexp 14138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-2nd 7430 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-n0 11620 df-z 11706 df-uz 11970 df-seq 13097 df-relexp 14139 |
This theorem is referenced by: relexpsucrd 14148 |
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