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Mirrors > Home > HSE Home > Th. List > hoadd32i | Structured version Visualization version GIF version |
Description: Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hods.1 | ⊢ 𝑅: ℋ⟶ ℋ |
hods.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hods.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoadd32i | ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hods.2 | . . . 4 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | hods.3 | . . . 4 ⊢ 𝑇: ℋ⟶ ℋ | |
3 | 1, 2 | hoaddcomi 31003 | . . 3 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) |
4 | 3 | oveq2i 7415 | . 2 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑅 +op (𝑇 +op 𝑆)) |
5 | hods.1 | . . 3 ⊢ 𝑅: ℋ⟶ ℋ | |
6 | 5, 1, 2 | hoaddassi 31007 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)) |
7 | 5, 2, 1 | hoaddassi 31007 | . 2 ⊢ ((𝑅 +op 𝑇) +op 𝑆) = (𝑅 +op (𝑇 +op 𝑆)) |
8 | 4, 6, 7 | 3eqtr4i 2771 | 1 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑅 +op 𝑇) +op 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⟶wf 6536 (class class class)co 7404 ℋchba 30150 +op chos 30169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-hilex 30230 ax-hfvadd 30231 ax-hvcom 30232 ax-hvass 30233 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8818 df-hosum 30961 |
This theorem is referenced by: hosubeq0i 31057 |
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