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Mirrors > Home > MPE Home > Th. List > Mathboxes > addcomgi | Structured version Visualization version GIF version |
Description: Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
addcomgi | β’ (π΄ + π΅) = (π΅ + π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcom 11424 | . 2 β’ ((π΄ β β β§ π΅ β β) β (π΄ + π΅) = (π΅ + π΄)) | |
2 | ax-addf 11211 | . . . 4 β’ + :(β Γ β)βΆβ | |
3 | 2 | fdmi 6728 | . . 3 β’ dom + = (β Γ β) |
4 | 3 | ndmovcom 7602 | . 2 β’ (Β¬ (π΄ β β β§ π΅ β β) β (π΄ + π΅) = (π΅ + π΄)) |
5 | 1, 4 | pm2.61i 182 | 1 β’ (π΄ + π΅) = (π΅ + π΄) |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1534 β wcel 2099 Γ cxp 5670 (class class class)co 7414 βcc 11130 + caddc 11135 |
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 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-addf 11211 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 |
This theorem is referenced by: addrcom 43906 |
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