hvsubsub4i - Hilbert Space Explorer

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Theorem hvsubsub4i 27300

Description: Hilbert vector space addition law. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hvass.1	⊢ 𝐴 ∈ ℋ
hvass.2	⊢ 𝐵 ∈ ℋ
hvass.3	⊢ 𝐶 ∈ ℋ
hvadd4.4	⊢ 𝐷 ∈ ℋ

**Assertion**
Ref	Expression
hvsubsub4i	⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

**Proof of Theorem hvsubsub4i**
Step	Hyp	Ref	Expression
1		hvass.1	. . . . 5 ⊢ 𝐴 ∈ ℋ
2		neg1cn 11001	. . . . . 6 ⊢ -1 ∈ ℂ
3		hvass.2	. . . . . 6 ⊢ 𝐵 ∈ ℋ
4	2, 3	hvmulcli 27255	. . . . 5 ⊢ (-1 ·_ℎ 𝐵) ∈ ℋ
5		hvass.3	. . . . . 6 ⊢ 𝐶 ∈ ℋ
6	2, 5	hvmulcli 27255	. . . . 5 ⊢ (-1 ·_ℎ 𝐶) ∈ ℋ
7		hvadd4.4	. . . . . . 7 ⊢ 𝐷 ∈ ℋ
8	2, 7	hvmulcli 27255	. . . . . 6 ⊢ (-1 ·_ℎ 𝐷) ∈ ℋ
9	2, 8	hvmulcli 27255	. . . . 5 ⊢ (-1 ·_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
10	1, 4, 6, 9	hvadd4i 27299	. . . 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 27292	. . . . 5 ⊢ (-1 ·_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐶) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
12	11	oveq2i 6560	. . . 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 27292	. . . . 5 ⊢ (-1 ·_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷))) = ((-1 ·_ℎ 𝐵) +_ℎ (-1 ·_ℎ (-1 ·_ℎ 𝐷)))
14	13	oveq2i 6560	. . . 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 2642	. . 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 27259	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐵)) ∈ ℋ
17	5, 8	hvaddcli 27259	. . . 4 ⊢ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
18	16, 17	hvsubvali 27261	. . 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 27259	. . . 4 ⊢ (𝐴 +_ℎ (-1 ·_ℎ 𝐶)) ∈ ℋ
20	3, 8	hvaddcli 27259	. . . 4 ⊢ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)) ∈ ℋ
21	19, 20	hvsubvali 27261	. . 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 2642	. 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 27261	. . 3 ⊢ (𝐴 −_ℎ 𝐵) = (𝐴 +_ℎ (-1 ·_ℎ 𝐵))
24	5, 7	hvsubvali 27261	. . 3 ⊢ (𝐶 −_ℎ 𝐷) = (𝐶 +_ℎ (-1 ·_ℎ 𝐷))
25	23, 24	oveq12i 6561	. 2 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐵)) −_ℎ (𝐶 +_ℎ (-1 ·_ℎ 𝐷)))
26	1, 5	hvsubvali 27261	. . 3 ⊢ (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶))
27	3, 7	hvsubvali 27261	. . 3 ⊢ (𝐵 −_ℎ 𝐷) = (𝐵 +_ℎ (-1 ·_ℎ 𝐷))
28	26, 27	oveq12i 6561	. 2 ⊢ ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷)) = ((𝐴 +_ℎ (-1 ·_ℎ 𝐶)) −_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐷)))
29	22, 25, 28	3eqtr4i 2642	1 ⊢ ((𝐴 −_ℎ 𝐵) −_ℎ (𝐶 −_ℎ 𝐷)) = ((𝐴 −_ℎ 𝐶) −_ℎ (𝐵 −_ℎ 𝐷))

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 ∈ wcel 1977 (class class class)co 6549 1c1 9816 -cneg 10146 ℋchil 27160 +_ℎ cva 27161 ·_ℎ csm 27162 −_ℎ cmv 27166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hfvmul 27246 ax-hvdistr1 27249

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 df-hvsub 27212

This theorem is referenced by: hvsubsub4 27301 pjsslem 27922

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