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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 30990 | . . 3 ⊢ (𝑅 +op 𝑆) = (𝑆 +op 𝑅) |
4 | 3 | oveq1i 7406 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = ((𝑆 +op 𝑅) +op 𝑇) |
5 | hods.3 | . . 3 ⊢ 𝑇: ℋ⟶ ℋ | |
6 | 1, 2, 5 | hoaddassi 30994 | . 2 ⊢ ((𝑅 +op 𝑆) +op 𝑇) = (𝑅 +op (𝑆 +op 𝑇)) |
7 | 2, 1, 5 | hoaddassi 30994 | . 2 ⊢ ((𝑆 +op 𝑅) +op 𝑇) = (𝑆 +op (𝑅 +op 𝑇)) |
8 | 4, 6, 7 | 3eqtr3i 2769 | 1 ⊢ (𝑅 +op (𝑆 +op 𝑇)) = (𝑆 +op (𝑅 +op 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⟶wf 6531 (class class class)co 7396 ℋchba 30137 +op chos 30156 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-hilex 30217 ax-hfvadd 30218 ax-hvcom 30219 ax-hvass 30220 |
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 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-map 8810 df-hosum 30948 |
This theorem is referenced by: ho0subi 31013 |
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