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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.) |
Ref | Expression |
---|---|
fovcl.1 | ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 |
Ref | Expression |
---|---|
fovcl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fovcl.1 | . . 3 ⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 | |
2 | ffnov 7278 | . . . 4 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 ↔ (𝐹 Fn (𝑅 × 𝑆) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶)) | |
3 | 2 | simprbi 499 | . . 3 ⊢ (𝐹:(𝑅 × 𝑆)⟶𝐶 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶 |
5 | oveq1 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | |
6 | 5 | eleq1d 2897 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝑦) ∈ 𝐶)) |
7 | oveq2 7164 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | |
8 | 7 | eleq1d 2897 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) ∈ 𝐶 ↔ (𝐴𝐹𝐵) ∈ 𝐶)) |
9 | 6, 8 | rspc2v 3633 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥𝐹𝑦) ∈ 𝐶 → (𝐴𝐹𝐵) ∈ 𝐶)) |
10 | 4, 9 | mpi 20 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 × cxp 5553 Fn wfn 6350 ⟶wf 6351 (class class class)co 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 |
This theorem is referenced by: addclnq 10367 mulclnq 10369 adderpq 10378 mulerpq 10379 distrnq 10383 axaddcl 10573 axmulcl 10575 xaddcl 12633 xmulcl 12667 elfzoelz 13039 addcnlem 23472 sgmcl 25723 hvaddcl 28789 hvmulcl 28790 hicl 28857 hhssabloilem 29038 rmxynorm 39535 rmxyneg 39537 rmxy1 39539 rmxy0 39540 rmxp1 39549 rmyp1 39550 rmxm1 39551 rmym1 39552 rmxluc 39553 rmyluc 39554 rmyluc2 39555 rmxdbl 39556 rmydbl 39557 rmxypos 39564 ltrmynn0 39565 ltrmxnn0 39566 lermxnn0 39567 rmxnn 39568 ltrmy 39569 rmyeq0 39570 rmyeq 39571 lermy 39572 rmynn 39573 rmynn0 39574 rmyabs 39575 jm2.24nn 39576 jm2.17a 39577 jm2.17b 39578 jm2.17c 39579 jm2.24 39580 rmygeid 39581 jm2.18 39605 jm2.19lem1 39606 jm2.19lem2 39607 jm2.19 39610 jm2.22 39612 jm2.23 39613 jm2.20nn 39614 jm2.25 39616 jm2.26a 39617 jm2.26lem3 39618 jm2.26 39619 jm2.15nn0 39620 jm2.16nn0 39621 jm2.27a 39622 jm2.27c 39624 rmydioph 39631 rmxdiophlem 39632 jm3.1lem1 39634 jm3.1 39637 expdiophlem1 39638 |
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