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Mirrors > Home > MPE Home > Th. List > 0oo | Structured version Visualization version GIF version |
Description: The zero operator is an operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0oo.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
0oo.2 | ⊢ 𝑌 = (BaseSet‘𝑊) |
0oo.0 | ⊢ 𝑍 = (𝑈 0op 𝑊) |
Ref | Expression |
---|---|
0oo | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6201 | . . . . 5 ⊢ (0vec‘𝑊) ∈ V | |
2 | 1 | fconst 6091 | . . . 4 ⊢ (𝑋 × {(0vec‘𝑊)}):𝑋⟶{(0vec‘𝑊)} |
3 | 0oo.2 | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
4 | eqid 2622 | . . . . . 6 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 27489 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ 𝑌) |
6 | 5 | snssd 4340 | . . . 4 ⊢ (𝑊 ∈ NrmCVec → {(0vec‘𝑊)} ⊆ 𝑌) |
7 | fss 6056 | . . . 4 ⊢ (((𝑋 × {(0vec‘𝑊)}):𝑋⟶{(0vec‘𝑊)} ∧ {(0vec‘𝑊)} ⊆ 𝑌) → (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌) | |
8 | 2, 6, 7 | sylancr 695 | . . 3 ⊢ (𝑊 ∈ NrmCVec → (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌) |
10 | 0oo.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
11 | 0oo.0 | . . . 4 ⊢ 𝑍 = (𝑈 0op 𝑊) | |
12 | 10, 4, 11 | 0ofval 27642 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 = (𝑋 × {(0vec‘𝑊)})) |
13 | 12 | feq1d 6030 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑍:𝑋⟶𝑌 ↔ (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌)) |
14 | 9, 13 | mpbird 247 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 NrmCVeccnv 27439 BaseSetcba 27441 0veccn0v 27443 0op c0o 27598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-grpo 27347 df-gid 27348 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 df-0o 27602 |
This theorem is referenced by: 0lno 27645 nmoo0 27646 nmlno0lem 27648 |
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