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| Mirrors > Home > MPE Home > Th. List > fovcl | Structured version Visualization version GIF version | ||
| Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.) |
| Ref | Expression |
|---|---|
| fovcl.1 | ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 |
| Ref | Expression |
|---|---|
| fovcl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fovcl.1 | . . . 4 ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 | |
| 2 | 1 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑅 → 𝐹:(𝑅 × 𝑆)⟶𝐶) |
| 3 | 2 | fovcld 7560 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
| 4 | 3 | 3anidm12 1421 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 × cxp 5683 ⟶wf 6557 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: addclnq 10985 mulclnq 10987 adderpq 10996 mulerpq 10997 distrnq 11001 axaddcl 11191 axmulcl 11193 xaddcl 13281 xmulcl 13315 elfzoelz 13699 cncrng 21401 addcnlem 24886 sgmcl 27189 hvaddcl 31031 hvmulcl 31032 hicl 31099 hhssabloilem 31280 rmxynorm 42930 rmxyneg 42932 rmxy1 42934 rmxy0 42935 rmxp1 42944 rmyp1 42945 rmxm1 42946 rmym1 42947 rmxluc 42948 rmyluc 42949 rmyluc2 42950 rmxdbl 42951 rmydbl 42952 rmxypos 42959 ltrmynn0 42960 ltrmxnn0 42961 lermxnn0 42962 rmxnn 42963 ltrmy 42964 rmyeq0 42965 rmyeq 42966 lermy 42967 rmynn 42968 rmynn0 42969 rmyabs 42970 jm2.24nn 42971 jm2.17a 42972 jm2.17b 42973 jm2.17c 42974 jm2.24 42975 rmygeid 42976 jm2.18 43000 jm2.19lem1 43001 jm2.19lem2 43002 jm2.19 43005 jm2.22 43007 jm2.23 43008 jm2.20nn 43009 jm2.25 43011 jm2.26a 43012 jm2.26lem3 43013 jm2.26 43014 jm2.15nn0 43015 jm2.16nn0 43016 jm2.27a 43017 jm2.27c 43019 rmydioph 43026 rmxdiophlem 43027 jm3.1lem1 43029 jm3.1 43032 expdiophlem1 43033 |
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