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Mirrors > Home > HSE Home > Th. List > hoadd12i | Structured version Visualization version GIF version |
Description: Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hods.1 | ⊢ 𝑅: ℋ⟶ ℋ |
hods.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hods.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoadd12i | ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hods.1 | . . . 4 ⊢ 𝑅: ℋ⟶ ℋ | |
2 | hods.2 | . . . 4 ⊢ 𝑆: ℋ⟶ ℋ | |
3 | 1, 2 | hoaddcomi 29549 | . . 3 ⊢ (𝑅 +op 𝑆) = (𝑆 +op 𝑅) |
4 | 3 | oveq1i 7166 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑆 +op 𝑅) +op 𝑇) |
5 | hods.3 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 1, 2, 5 | hoaddassi 29553 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)) |
7 | 2, 1, 5 | hoaddassi 29553 | . 2 ⊢ ((𝑆 +op 𝑅) +op 𝑇) = (𝑆 +op (𝑅 +op 𝑇)) |
8 | 4, 6, 7 | 3eqtr3i 2852 | 1 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⟶wf 6351 (class class class)co 7156 ℋchba 28696 +op chos 28715 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-hilex 28776 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 |
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-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-hosum 29507 |
This theorem is referenced by: ho0subi 29572 |
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