Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elimnv | Structured version Visualization version GIF version |
Description: Hypothesis elimination lemma for normed complex vector spaces to assist weak deduction theorem. (Contributed by NM, 16-May-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
elimnv.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
elimnv.5 | ⊢ 𝑍 = (0vec‘𝑈) |
elimnv.9 | ⊢ 𝑈 ∈ NrmCVec |
Ref | Expression |
---|---|
elimnv | ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elimnv.9 | . . 3 ⊢ 𝑈 ∈ NrmCVec | |
2 | elimnv.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | elimnv.5 | . . . 4 ⊢ 𝑍 = (0vec‘𝑈) | |
4 | 2, 3 | nvzcl 28405 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 𝑍 ∈ 𝑋 |
6 | 5 | elimel 4533 | 1 ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ifcif 4466 ‘cfv 6349 NrmCVeccnv 28355 BaseSetcba 28357 0veccn0v 28359 |
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-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-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 |
This theorem is referenced by: elimph 28591 |
Copyright terms: Public domain | W3C validator |