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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 12380 | . . . . . 6 ⊢ -1 ∈ ℂ | |
| 3 | hvass.2 | . . . . . 6 ⊢ 𝐵 ∈ ℋ | |
| 4 | 2, 3 | hvmulcli 31033 | . . . . 5 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ | 
| 5 | hvass.3 | . . . . . 6 ⊢ 𝐶 ∈ ℋ | |
| 6 | 2, 5 | hvmulcli 31033 | . . . . 5 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ | 
| 7 | hvadd4.4 | . . . . . . 7 ⊢ 𝐷 ∈ ℋ | |
| 8 | 2, 7 | hvmulcli 31033 | . . . . . 6 ⊢ (-1 ·ℎ 𝐷) ∈ ℋ | 
| 9 | 2, 8 | hvmulcli 31033 | . . . . 5 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐷)) ∈ ℋ | 
| 10 | 1, 4, 6, 9 | hvadd4i 31077 | . . . 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 31070 | . . . . 5 ⊢ (-1 ·ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) = ((-1 ·ℎ 𝐶) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷))) | 
| 12 | 11 | oveq2i 7442 | . . . 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 31070 | . . . . 5 ⊢ (-1 ·ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐷))) | 
| 14 | 13 | oveq2i 7442 | . . . 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 2775 | . . 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 31037 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ | 
| 17 | 5, 8 | hvaddcli 31037 | . . . 4 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐷)) ∈ ℋ | 
| 18 | 16, 17 | hvsubvali 31039 | . . 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 31037 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐶)) ∈ ℋ | 
| 20 | 3, 8 | hvaddcli 31037 | . . . 4 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐷)) ∈ ℋ | 
| 21 | 19, 20 | hvsubvali 31039 | . . 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 2775 | . 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 31039 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) | 
| 24 | 5, 7 | hvsubvali 31039 | . . 3 ⊢ (𝐶 −ℎ 𝐷) = (𝐶 +ℎ (-1 ·ℎ 𝐷)) | 
| 25 | 23, 24 | oveq12i 7443 | . 2 ⊢ ((𝐴 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ (𝐶 +ℎ (-1 ·ℎ 𝐷))) | 
| 26 | 1, 5 | hvsubvali 31039 | . . 3 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) | 
| 27 | 3, 7 | hvsubvali 31039 | . . 3 ⊢ (𝐵 −ℎ 𝐷) = (𝐵 +ℎ (-1 ·ℎ 𝐷)) | 
| 28 | 26, 27 | oveq12i 7443 | . 2 ⊢ ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷)) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) −ℎ (𝐵 +ℎ (-1 ·ℎ 𝐷))) | 
| 29 | 22, 25, 28 | 3eqtr4i 2775 | 1 ⊢ ((𝐴 −ℎ 𝐵) −ℎ (𝐶 −ℎ 𝐷)) = ((𝐴 −ℎ 𝐶) −ℎ (𝐵 −ℎ 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 (class class class)co 7431 1c1 11156 -cneg 11493 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 −ℎ cmv 30944 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hfvmul 31024 ax-hvdistr1 31027 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 df-hvsub 30990 | 
| This theorem is referenced by: hvsubsub4 31079 pjsslem 31698 | 
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