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Mirrors > Home > HSE Home > Th. List > hvsubsub4i | Structured version Visualization version GIF version |
Description: Hilbert vector space addition law. (Contributed by NM, 31-Oct-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 |
---|---|
hvsubsub4i | ⊢ ((𝐴 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvass.1 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
2 | neg1cn 11745 | . . . . . 6 ⊢ -1 ∈ ℂ | |
3 | hvass.2 | . . . . . 6 ⊢ 𝐵 ∈ ℋ | |
4 | 2, 3 | hvmulcli 28785 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
5 | hvass.3 | . . . . . 6 ⊢ 𝐶 ∈ ℋ | |
6 | 2, 5 | hvmulcli 28785 | . . . . 5 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ |
7 | hvadd4.4 | . . . . . . 7 ⊢ 𝐷 ∈ ℋ | |
8 | 2, 7 | hvmulcli 28785 | . . . . . 6 ⊢ (-1 ·ℎ 𝐷) ∈ ℋ |
9 | 2, 8 | hvmulcli 28785 | . . . . 5 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐷)) ∈ ℋ |
10 | 1, 4, 6, 9 | hvadd4i 28829 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ ((-1 ·ℎ 𝐶) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷)))) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷)))) |
11 | 2, 5, 8 | hvdistr1i 28822 | . . . . 5 ⊢ (-1 ·ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) = ((-1 ·ℎ 𝐶) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷))) |
12 | 11 | oveq2i 7161 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷)))) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ ((-1 ·ℎ 𝐶) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷)))) |
13 | 2, 3, 8 | hvdistr1i 28822 | . . . . 5 ⊢ (-1 ·ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷))) |
14 | 13 | oveq2i 7161 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (-1 ·ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷)))) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷)))) |
15 | 10, 12, 14 | 3eqtr4i 2854 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷)))) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (-1 ·ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷)))) |
16 | 1, 4 | hvaddcli 28789 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
17 | 5, 8 | hvaddcli 28789 | . . . 4 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐷)) ∈ ℋ |
18 | 16, 17 | hvsubvali 28791 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷)))) |
19 | 1, 6 | hvaddcli 28789 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐶)) ∈ ℋ |
20 | 3, 8 | hvaddcli 28789 | . . . 4 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐷)) ∈ ℋ |
21 | 19, 20 | hvsubvali 28791 | . . 3 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐶)) −ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (-1 ·ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷)))) |
22 | 15, 18, 21 | 3eqtr4i 2854 | . 2 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) −ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) |
23 | 1, 3 | hvsubvali 28791 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
24 | 5, 7 | hvsubvali 28791 | . . 3 ⊢ (𝐶 −ℎ 𝐷) = (𝐶 +ℎ (-1 ·ℎ 𝐷)) |
25 | 23, 24 | oveq12i 7162 | . 2 ⊢ ((𝐴 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) |
26 | 1, 5 | hvsubvali 28791 | . . 3 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) |
27 | 3, 7 | hvsubvali 28791 | . . 3 ⊢ (𝐵 −ℎ 𝐷) = (𝐵 +ℎ (-1 ·ℎ 𝐷)) |
28 | 26, 27 | oveq12i 7162 | . 2 ⊢ ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷)) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) −ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) |
29 | 22, 25, 28 | 3eqtr4i 2854 | 1 ⊢ ((𝐴 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7150 1c1 10532 -cneg 10865 ℋchba 28690 +ℎ cva 28691 ·ℎ csm 28692 −ℎ cmv 28696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hfvmul 28776 ax-hvdistr1 28779 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-hvsub 28742 |
This theorem is referenced by: hvsubsub4 28831 pjsslem 29450 |
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