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Mirrors > Home > MPE Home > Th. List > adds32d | Structured version Visualization version GIF version |
Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Scott Fenton, 22-Jan-2025.) |
Ref | Expression |
---|---|
addsassd.1 | ⊢ (𝜑 → 𝐴 ∈ No ) |
addsassd.2 | ⊢ (𝜑 → 𝐵 ∈ No ) |
addsassd.3 | ⊢ (𝜑 → 𝐶 ∈ No ) |
Ref | Expression |
---|---|
adds32d | ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((𝐴 +s 𝐶) +s 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsassd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ No ) | |
2 | addsassd.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ No ) | |
3 | 1, 2 | addscomd 27902 | . . 3 ⊢ (𝜑 → (𝐵 +s 𝐶) = (𝐶 +s 𝐵)) |
4 | 3 | oveq2d 7440 | . 2 ⊢ (𝜑 → (𝐴 +s (𝐵 +s 𝐶)) = (𝐴 +s (𝐶 +s 𝐵))) |
5 | addsassd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ No ) | |
6 | 5, 1, 2 | addsassd 27941 | . 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = (𝐴 +s (𝐵 +s 𝐶))) |
7 | 5, 2, 1 | addsassd 27941 | . 2 ⊢ (𝜑 → ((𝐴 +s 𝐶) +s 𝐵) = (𝐴 +s (𝐶 +s 𝐵))) |
8 | 4, 6, 7 | 3eqtr4d 2777 | 1 ⊢ (𝜑 → ((𝐴 +s 𝐵) +s 𝐶) = ((𝐴 +s 𝐶) +s 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7424 No csur 27591 +s cadds 27894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-1o 8491 df-2o 8492 df-nadd 8691 df-no 27594 df-slt 27595 df-bday 27596 df-sle 27696 df-sslt 27732 df-scut 27734 df-0s 27775 df-made 27792 df-old 27793 df-left 27795 df-right 27796 df-norec2 27884 df-adds 27895 |
This theorem is referenced by: adds4d 27944 subadds 27996 addsubsd 28008 sltsubsubbd 28009 readdscl 28245 |
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