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Mirrors > Home > ILE Home > Th. List > lmod4 | GIF version |
Description: Commutative/associative law for left module vector sum. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lmod4.v | ⊢ 𝑉 = (Base‘𝑊) |
lmod4.p | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
lmod4 | ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmodcmn 13611 | . 2 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ CMnd) | |
2 | lmod4.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | lmod4.p | . . 3 ⊢ + = (+g‘𝑊) | |
4 | 2, 3 | cmn4 13204 | . 2 ⊢ ((𝑊 ∈ CMnd ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈))) |
5 | 1, 4 | syl3an1 1281 | 1 ⊢ ((𝑊 ∈ LMod ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ (𝑍 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉)) → ((𝑋 + 𝑌) + (𝑍 + 𝑈)) = ((𝑋 + 𝑍) + (𝑌 + 𝑈))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 979 = wceq 1363 ∈ wcel 2159 ‘cfv 5230 (class class class)co 5890 Basecbs 12479 +gcplusg 12554 CMndccmn 13183 LModclmod 13563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-coll 4132 ax-sep 4135 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-cnex 7919 ax-resscn 7920 ax-1cn 7921 ax-1re 7922 ax-icn 7923 ax-addcl 7924 ax-addrcl 7925 ax-mulcl 7926 ax-addcom 7928 ax-addass 7930 ax-i2m1 7933 ax-0lt1 7934 ax-0id 7936 ax-rnegex 7937 ax-pre-ltirr 7940 ax-pre-ltadd 7944 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-nel 2455 df-ral 2472 df-rex 2473 df-reu 2474 df-rmo 2475 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-id 4307 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-riota 5846 df-ov 5893 df-oprab 5894 df-mpo 5895 df-pnf 8011 df-mnf 8012 df-ltxr 8014 df-inn 8937 df-2 8995 df-3 8996 df-4 8997 df-5 8998 df-6 8999 df-ndx 12482 df-slot 12483 df-base 12485 df-sets 12486 df-plusg 12567 df-mulr 12568 df-sca 12570 df-vsca 12571 df-0g 12728 df-mgm 12797 df-sgrp 12830 df-mnd 12843 df-grp 12913 df-minusg 12914 df-cmn 13185 df-abl 13186 df-mgp 13235 df-ur 13274 df-ring 13312 df-lmod 13565 |
This theorem is referenced by: (None) |
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