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Mirrors > Home > HSE Home > Th. List > hvadd4i | Structured version Visualization version GIF version |
Description: Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvass.1 | ⊢ 𝐴 ∈ ℋ |
hvass.2 | ⊢ 𝐵 ∈ ℋ |
hvass.3 | ⊢ 𝐶 ∈ ℋ |
hvadd4.4 | ⊢ 𝐷 ∈ ℋ |
Ref | Expression |
---|---|
hvadd4i | ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvass.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | hvass.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | hvass.3 | . 2 ⊢ 𝐶 ∈ ℋ | |
4 | hvadd4.4 | . 2 ⊢ 𝐷 ∈ ℋ | |
5 | hvadd4 28819 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷))) | |
6 | 1, 2, 3, 4, 5 | mp4an 692 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℋchba 28702 +ℎ cva 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 |
This theorem is referenced by: hvsubsub4i 28842 hvsubcan2i 28847 pjaddii 29458 |
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