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 28413 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑍 ∈ 𝑋) |
5 | 1, 4 | ax-mp 5 | . 2 ⊢ 𝑍 ∈ 𝑋 |
6 | 5 | elimel 4536 | 1 ⊢ if(𝐴 ∈ 𝑋, 𝐴, 𝑍) ∈ 𝑋 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ifcif 4469 ‘cfv 6357 NrmCVeccnv 28363 BaseSetcba 28365 0veccn0v 28367 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-1st 7691 df-2nd 7692 df-grpo 28272 df-gid 28273 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-nmcv 28379 |
This theorem is referenced by: elimph 28599 |
Copyright terms: Public domain | W3C validator |