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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 28051 | . . . . 5 ⊢ (𝜑 → (𝐴 +s ( -us ‘𝐴)) = 0s ) |
3 | 2 | oveq1d 7439 | . . . 4 ⊢ (𝜑 → ((𝐴 +s ( -us ‘𝐴)) ·s 𝐵) = ( 0s ·s 𝐵)) |
4 | 1 | negscld 28046 | . . . . 5 ⊢ (𝜑 → ( -us ‘𝐴) ∈ No ) |
5 | mulnegs1d.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ No ) | |
6 | 1, 4, 5 | addsdird 28158 | . . . 4 ⊢ (𝜑 → ((𝐴 +s ( -us ‘𝐴)) ·s 𝐵) = ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵))) |
7 | muls02 28142 | . . . . 5 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s ) | |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s ) |
9 | 3, 6, 8 | 3eqtr3d 2774 | . . 3 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵)) = 0s ) |
10 | 1, 5 | mulscld 28136 | . . . 4 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No ) |
11 | 10 | negsidd 28051 | . . 3 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s ( -us ‘(𝐴 ·s 𝐵))) = 0s ) |
12 | 9, 11 | eqtr4d 2769 | . 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) +s (( -us ‘𝐴) ·s 𝐵)) = ((𝐴 ·s 𝐵) +s ( -us ‘(𝐴 ·s 𝐵)))) |
13 | 4, 5 | mulscld 28136 | . . 3 ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) ∈ No ) |
14 | 10 | negscld 28046 | . . 3 ⊢ (𝜑 → ( -us ‘(𝐴 ·s 𝐵)) ∈ No ) |
15 | 13, 14, 10 | addscan1d 28014 | . 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 231 | 1 ⊢ (𝜑 → (( -us ‘𝐴) ·s 𝐵) = ( -us ‘(𝐴 ·s 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 No csur 27669 0s c0s 27852 +s cadds 27973 -us cnegs 28029 ·s cmuls 28107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-ot 4642 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-1o 8496 df-2o 8497 df-nadd 8696 df-no 27672 df-slt 27673 df-bday 27674 df-sle 27775 df-sslt 27811 df-scut 27813 df-0s 27854 df-made 27871 df-old 27872 df-left 27874 df-right 27875 df-norec 27952 df-norec2 27963 df-adds 27974 df-negs 28031 df-subs 28032 df-muls 28108 |
This theorem is referenced by: mulnegs2d 28162 mul2negsd 28163 precsexlem9 28214 recsex 28218 absmuls 28239 |
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