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Mirrors > Home > HSE Home > Th. List > hvsubadd | Structured version Visualization version GIF version |
Description: Relationship between vector subtraction and addition. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubadd | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6981 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (𝐴 −ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) | |
2 | 1 | eqeq1d 2774 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = 𝐶)) |
3 | eqeq2 2783 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((𝐵 +ℎ 𝐶) = 𝐴 ↔ (𝐵 +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) | |
4 | 2, 3 | bibi12d 338 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))) |
5 | oveq2 6982 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
6 | 5 | eqeq1d 2774 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = 𝐶 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = 𝐶)) |
7 | oveq1 6981 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (𝐵 +ℎ 𝐶) = (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶)) | |
8 | 7 | eqeq1d 2774 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((𝐵 +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) |
9 | 6, 8 | bibi12d 338 | . 2 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = 𝐶 ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))) |
10 | eqeq2 2783 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = 𝐶 ↔ (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) | |
11 | oveq2 6982 | . . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶) = (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ))) | |
12 | 11 | eqeq1d 2774 | . . 3 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → ((if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) |
13 | 10, 12 | bibi12d 338 | . 2 ⊢ (𝐶 = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) → (((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = 𝐶 ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ 𝐶) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) ↔ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)))) |
14 | ifhvhv0 28590 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
15 | ifhvhv0 28590 | . . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
16 | ifhvhv0 28590 | . . 3 ⊢ if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ∈ ℋ | |
17 | 14, 15, 16 | hvsubaddi 28634 | . 2 ⊢ ((if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)) = if(𝐶 ∈ ℋ, 𝐶, 0ℎ) ↔ (if(𝐵 ∈ ℋ, 𝐵, 0ℎ) +ℎ if(𝐶 ∈ ℋ, 𝐶, 0ℎ)) = if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) |
18 | 4, 9, 13, 17 | dedth3h 4402 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) = 𝐶 ↔ (𝐵 +ℎ 𝐶) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ifcif 4344 (class class class)co 6974 ℋchba 28487 +ℎ cva 28488 0ℎc0v 28492 −ℎ cmv 28493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-hfvadd 28568 ax-hvcom 28569 ax-hvass 28570 ax-hv0cl 28571 ax-hvaddid 28572 ax-hfvmul 28573 ax-hvmulid 28574 ax-hvdistr2 28577 ax-hvmul0 28578 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-sub 10670 df-neg 10671 df-hvsub 28539 |
This theorem is referenced by: shmodsi 28959 pjop 28997 pjpo 28998 chscllem2 29208 pjo 29241 hodsi 29345 pjimai 29746 superpos 29924 sumdmdii 29985 sumdmdlem 29988 |
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