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Mirrors > Home > HSE Home > Th. List > bra0 | Structured version Visualization version GIF version |
Description: The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bra0 | ⊢ (bra‘0ℎ) = ( ℋ × {0}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 30928 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | brafval 31868 | . . . 4 ⊢ (0ℎ ∈ ℋ → (bra‘0ℎ) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 0ℎ))) | |
3 | hi02 31022 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑥 ·ih 0ℎ) = 0) | |
4 | 3 | mpteq2ia 5255 | . . . 4 ⊢ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 0ℎ)) = (𝑥 ∈ ℋ ↦ 0) |
5 | 2, 4 | eqtrdi 2781 | . . 3 ⊢ (0ℎ ∈ ℋ → (bra‘0ℎ) = (𝑥 ∈ ℋ ↦ 0)) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (bra‘0ℎ) = (𝑥 ∈ ℋ ↦ 0) |
7 | fconstmpt 5743 | . 2 ⊢ ( ℋ × {0}) = (𝑥 ∈ ℋ ↦ 0) | |
8 | 6, 7 | eqtr4i 2756 | 1 ⊢ (bra‘0ℎ) = ( ℋ × {0}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {csn 4632 ↦ cmpt 5235 × cxp 5679 ‘cfv 6553 (class class class)co 7423 0cc0 11154 ℋchba 30844 ·ih csp 30847 0ℎc0v 30849 bracbr 30881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-hilex 30924 ax-hv0cl 30928 ax-hvmul0 30935 ax-hfi 31004 ax-his1 31007 ax-his3 31009 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-2 12322 df-cj 15099 df-re 15100 df-im 15101 df-bra 31775 |
This theorem is referenced by: branmfn 32030 |
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