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Mirrors > Home > HSE Home > Th. List > hvaddsub12 | Structured version Visualization version GIF version |
Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddsub12 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 −ℎ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11754 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | hvmulcl 28793 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ 𝐶) ∈ ℋ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐶 ∈ ℋ → (-1 ·ℎ 𝐶) ∈ ℋ) |
4 | hvadd12 28815 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (-1 ·ℎ 𝐶) ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶))) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) | |
5 | 3, 4 | syl3an3 1161 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶))) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
6 | hvsubval 28796 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −ℎ 𝐶) = (𝐵 +ℎ (-1 ·ℎ 𝐶))) | |
7 | 6 | oveq2d 7175 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶)))) |
8 | 7 | 3adant1 1126 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶)))) |
9 | hvsubval 28796 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶))) | |
10 | 9 | oveq2d 7175 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ (𝐴 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
11 | 10 | 3adant2 1127 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ (𝐴 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
12 | 5, 8, 11 | 3eqtr4d 2869 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 −ℎ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 (class class class)co 7159 ℂcc 10538 1c1 10541 -cneg 10874 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 −ℎ cmv 28705 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-hvcom 28781 ax-hvass 28782 ax-hfvmul 28785 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-ltxr 10683 df-sub 10875 df-neg 10876 df-hvsub 28751 |
This theorem is referenced by: 5oalem1 29434 3oalem2 29443 pjcji 29464 pjclem4 29979 pj3si 29987 |
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