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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 6677 | . . . . 5 ⊢ (0vec‘𝑊) ∈ V | |
2 | 1 | fconst 6559 | . . . 4 ⊢ (𝑋 × {(0vec‘𝑊)}):𝑋⟶{(0vec‘𝑊)} |
3 | 0oo.2 | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
4 | eqid 2821 | . . . . . 6 ⊢ (0vec‘𝑊) = (0vec‘𝑊) | |
5 | 3, 4 | nvzcl 28405 | . . . . 5 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ 𝑌) |
6 | 5 | snssd 4735 | . . . 4 ⊢ (𝑊 ∈ NrmCVec → {(0vec‘𝑊)} ⊆ 𝑌) |
7 | fss 6521 | . . . 4 ⊢ (((𝑋 × {(0vec‘𝑊)}):𝑋⟶{(0vec‘𝑊)} ∧ {(0vec‘𝑊)} ⊆ 𝑌) → (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌) | |
8 | 2, 6, 7 | sylancr 589 | . . 3 ⊢ (𝑊 ∈ NrmCVec → (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌) |
9 | 8 | adantl 484 | . 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 28558 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍 = (𝑋 × {(0vec‘𝑊)})) |
13 | 12 | feq1d 6493 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → (𝑍:𝑋⟶𝑌 ↔ (𝑋 × {(0vec‘𝑊)}):𝑋⟶𝑌)) |
14 | 9, 13 | mpbird 259 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝑍:𝑋⟶𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 {csn 4560 × cxp 5547 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 NrmCVeccnv 28355 BaseSetcba 28357 0veccn0v 28359 0op c0o 28514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-nmcv 28371 df-0o 28518 |
This theorem is referenced by: 0lno 28561 nmoo0 28562 nmlno0lem 28564 |
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