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Mirrors > Home > HSE Home > Th. List > hvsubaddi | Structured version Visualization version GIF version |
Description: Relationship between vector subtraction and addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
hvaddcan.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvsubaddi | ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 28797 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | eqeq1i 2826 | . 2 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
5 | neg1cn 11752 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28791 | . . . . . 6 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 2, 1, 6 | hvadd12i 28834 | . . . . 5 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) |
8 | 2 | hvnegidi 28807 | . . . . . 6 ⊢ (𝐵 +ℎ (-1 ·ℎ 𝐵)) = 0ℎ |
9 | 8 | oveq2i 7167 | . . . . 5 ⊢ (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ 0ℎ) |
10 | ax-hvaddid 28781 | . . . . . 6 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ (𝐴 +ℎ 0ℎ) = 𝐴 |
12 | 7, 9, 11 | 3eqtri 2848 | . . . 4 ⊢ (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = 𝐴 |
13 | 12 | eqeq1i 2826 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ 𝐴 = (𝐵 +ℎ 𝐶)) |
14 | 1, 6 | hvaddcli 28795 | . . . 4 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
15 | hvaddcan.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
16 | 2, 14, 15 | hvaddcani 28842 | . . 3 ⊢ ((𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = (𝐵 +ℎ 𝐶) ↔ (𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶) |
17 | eqcom 2828 | . . 3 ⊢ (𝐴 = (𝐵 +ℎ 𝐶) ↔ (𝐵 +ℎ 𝐶) = 𝐴) | |
18 | 13, 16, 17 | 3bitr3i 303 | . 2 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐵)) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
19 | 4, 18 | bitri 277 | 1 ⊢ ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 1c1 10538 -cneg 10871 ℋchba 28696 +ℎ cva 28697 ·ℎ csm 28698 0ℎc0v 28701 −ℎ cmv 28702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-hfvadd 28777 ax-hvcom 28778 ax-hvass 28779 ax-hv0cl 28780 ax-hvaddid 28781 ax-hfvmul 28782 ax-hvmulid 28783 ax-hvdistr2 28786 ax-hvmul0 28787 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-hvsub 28748 |
This theorem is referenced by: hvsubadd 28854 omlsilem 29179 |
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