Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > suppssr | Structured version Visualization version GIF version |
Description: A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) |
Ref | Expression |
---|---|
suppssr.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
suppssr.n | ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) |
suppssr.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
suppssr.z | ⊢ (𝜑 → 𝑍 ∈ 𝑈) |
Ref | Expression |
---|---|
suppssr | ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3550 | . 2 ⊢ (𝑋 ∈ (𝐴 ∖ 𝑊) ↔ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) | |
2 | fvex 6113 | . . . . . 6 ⊢ (𝐹‘𝑋) ∈ V | |
3 | eldifsn 4260 | . . . . . 6 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍)) | |
4 | 2, 3 | mpbiran 955 | . . . . 5 ⊢ ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) ↔ (𝐹‘𝑋) ≠ 𝑍) |
5 | suppssr.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
6 | ffn 5958 | . . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
7 | 5, 6 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
8 | suppssr.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
9 | suppssr.z | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝑈) | |
10 | elsuppfn 7190 | . . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑈) → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
11 | 7, 8, 9, 10 | syl3anc 1318 | . . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍))) |
12 | ibar 524 | . . . . . . . . . . 11 ⊢ ((𝐹‘𝑋) ∈ V → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))) | |
13 | 2, 12 | mp1i 13 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ ((𝐹‘𝑋) ∈ V ∧ (𝐹‘𝑋) ≠ 𝑍))) |
14 | 13, 3 | syl6bbr 277 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 ↔ (𝐹‘𝑋) ∈ (V ∖ {𝑍}))) |
15 | 14 | pm5.32da 671 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ≠ 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
16 | 11, 15 | bitrd 267 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) ↔ (𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})))) |
17 | suppssr.n | . . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊) | |
18 | 17 | sseld 3567 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (𝐹 supp 𝑍) → 𝑋 ∈ 𝑊)) |
19 | 16, 18 | sylbird 249 | . . . . . 6 ⊢ (𝜑 → ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ (V ∖ {𝑍})) → 𝑋 ∈ 𝑊)) |
20 | 19 | expdimp 452 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ∈ (V ∖ {𝑍}) → 𝑋 ∈ 𝑊)) |
21 | 4, 20 | syl5bir 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) ≠ 𝑍 → 𝑋 ∈ 𝑊)) |
22 | 21 | necon1bd 2800 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (¬ 𝑋 ∈ 𝑊 → (𝐹‘𝑋) = 𝑍)) |
23 | 22 | impr 647 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐴 ∧ ¬ 𝑋 ∈ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
24 | 1, 23 | sylan2b 491 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-supp 7183 |
This theorem is referenced by: fsuppmptif 8188 fsuppco2 8191 fsuppcor 8192 cantnfp1lem1 8458 cantnfp1lem3 8460 cantnflem1 8469 cnfcom2lem 8481 gsumval3 18131 gsumcllem 18132 gsumzaddlem 18144 gsumzmhm 18160 gsumpt 18184 gsum2dlem1 18192 gsum2dlem2 18193 gsum2d 18194 dprdfinv 18241 dprdfadd 18242 dmdprdsplitlem 18259 dpjidcl 18280 gsumdixp 18432 lcomfsupp 18726 psrbaglesupp 19189 psrbagaddcl 19191 psrbaglefi 19193 mplsubglem 19255 mpllsslem 19256 mplsubrglem 19260 mplmonmul 19285 mplcoe1 19286 mplcoe5 19289 mplbas2 19291 evlslem4 19329 evlslem2 19333 uvcresum 19951 frlmsslsp 19954 rrxcph 22988 rrxmval 22996 rrxmetlem 22998 rrxmet 22999 rrxdstprj1 23000 deg1mul3le 23680 eulerpartlemb 29757 |
Copyright terms: Public domain | W3C validator |