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| Mirrors > Home > MPE Home > Th. List > mulnegs1d | Structured version Visualization version GIF version | ||
| Description: Product with negative is negative of product. Part of theorem 7 of [Conway] p. 19. (Contributed by Scott Fenton, 10-Mar-2025.) |
| Ref | Expression |
|---|---|
| mulnegs1d.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
| mulnegs1d.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
| Ref | Expression |
|---|---|
| mulnegs1d | ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mulnegs1d.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No ) | |
| 2 | 1 | negsidd 28074 | . . . . 5 ⊢ (𝜑 → (𝐴 +s ( -us ‘𝐴)) = 0s ) |
| 3 | 2 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → ((𝐴 +s ( -us ‘𝐴)) ·s 𝐵) = ( 0s ·s 𝐵)) |
| 4 | 1 | negscld 28069 | . . . . 5 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No ) |
| 5 | mulnegs1d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ No ) | |
| 6 | 1, 4, 5 | addsdird 28183 | . . . 4 ⊢ (𝜑 → ((𝐴 +s ( -us ‘𝐴)) ·s 𝐵) = ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵))) |
| 7 | muls02 28167 | . . . . 5 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s ) | |
| 8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s ) |
| 9 | 3, 6, 8 | 3eqtr3d 2785 | . . 3 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵)) = 0s ) |
| 10 | 1, 5 | mulscld 28161 | . . . 4 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No ) |
| 11 | 10 | negsidd 28074 | . . 3 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s ( -us ‘(𝐴 ·s 𝐵))) = 0s ) |
| 12 | 9, 11 | eqtr4d 2780 | . 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ( -us ‘(𝐴 ·s 𝐵)))) |
| 13 | 4, 5 | mulscld 28161 | . . 3 ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) ∈ No ) |
| 14 | 10 | negscld 28069 | . . 3 ⊢ (𝜑 → ( -us ‘(𝐴 ·s 𝐵)) ∈ No ) |
| 15 | 13, 14, 10 | addscan1d 28033 | . 2 ⊢ (𝜑 → (((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ( -us ‘(𝐴 ·s 𝐵))) ↔ (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵)))) |
| 16 | 12, 15 | mpbid 232 | 1 ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 No csur 27684 0s c0s 27867 +s cadds 27992 -us cnegs 28051 ·s cmuls 28132 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-ral 3062 df-rex 3071 df-rmo 3380 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 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-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-subs 28054 df-muls 28133 |
| This theorem is referenced by: mulnegs2d 28187 mul2negsd 28188 precsexlem9 28239 recsex 28243 absmuls 28268 |
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