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Mirrors > Home > MPE Home > Th. List > relexpsucrd | Structured version Visualization version GIF version |
Description: A reduction for relation exponentiation to the right. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
relexpsucrd.1 | ⊢ (𝜑 → Rel 𝑅) |
relexpsucrd.2 | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
relexpsucrd | ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑅 ∈ V) | |
2 | relexpsucrd.1 | . . . . 5 ⊢ (𝜑 → Rel 𝑅) | |
3 | 2 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → Rel 𝑅) |
4 | relexpsucrd.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → 𝑁 ∈ ℕ0) |
6 | relexpsucr 14741 | . . . 4 ⊢ ((𝑅 ∈ V ∧ Rel 𝑅 ∧ 𝑁 ∈ ℕ0) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) | |
7 | 1, 3, 5, 6 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ 𝑅 ∈ V) → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
8 | 7 | ex 413 | . 2 ⊢ (𝜑 → (𝑅 ∈ V → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅))) |
9 | reldmrelexp 14730 | . . . 4 ⊢ Rel dom ↑𝑟 | |
10 | 9 | ovprc1 7316 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟(𝑁 + 1)) = ∅) |
11 | 9 | ovprc1 7316 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟𝑁) = ∅) |
12 | 11 | coeq1d 5772 | . . . 4 ⊢ (¬ 𝑅 ∈ V → ((𝑅↑𝑟𝑁) ∘ 𝑅) = (∅ ∘ 𝑅)) |
13 | co01 6167 | . . . 4 ⊢ (∅ ∘ 𝑅) = ∅ | |
14 | 12, 13 | eqtr2di 2795 | . . 3 ⊢ (¬ 𝑅 ∈ V → ∅ = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
15 | 10, 14 | eqtrd 2778 | . 2 ⊢ (¬ 𝑅 ∈ V → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
16 | 8, 15 | pm2.61d1 180 | 1 ⊢ (𝜑 → (𝑅↑𝑟(𝑁 + 1)) = ((𝑅↑𝑟𝑁) ∘ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3431 ∅c0 4258 ∘ ccom 5595 Rel wrel 5596 (class class class)co 7277 1c1 10870 + caddc 10872 ℕ0cn0 12231 ↑𝑟crelexp 14728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 df-seq 13720 df-relexp 14729 |
This theorem is referenced by: rtrclreclem4 14770 relexpiidm 41282 |
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