Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 28797 | . 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 691 | 1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (𝐶 +ℎ 𝐷)) = ((𝐴 +ℎ 𝐶) +ℎ (𝐵 +ℎ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7142 ℋchba 28680 +ℎ cva 28681 |
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 2793 ax-sep 5189 ax-nul 5196 ax-pr 5316 ax-hfvadd 28761 ax-hvcom 28762 ax-hvass 28763 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-ov 7145 |
This theorem is referenced by: hvsubsub4i 28820 hvsubcan2i 28825 pjaddii 29436 |
Copyright terms: Public domain | W3C validator |