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Theorem gsumwrev 17619
 Description: A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
gsumwrev.b 𝐵 = (Base‘𝑀)
gsumwrev.o 𝑂 = (oppg𝑀)
Assertion
Ref Expression
gsumwrev ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))

Proof of Theorem gsumwrev
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6557 . . . . 5 (𝑥 = ∅ → (𝑂 Σg 𝑥) = (𝑂 Σg ∅))
2 fveq2 6103 . . . . . . 7 (𝑥 = ∅ → (reverse‘𝑥) = (reverse‘∅))
3 rev0 13364 . . . . . . 7 (reverse‘∅) = ∅
42, 3syl6eq 2660 . . . . . 6 (𝑥 = ∅ → (reverse‘𝑥) = ∅)
54oveq2d 6565 . . . . 5 (𝑥 = ∅ → (𝑀 Σg (reverse‘𝑥)) = (𝑀 Σg ∅))
61, 5eqeq12d 2625 . . . 4 (𝑥 = ∅ → ((𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥)) ↔ (𝑂 Σg ∅) = (𝑀 Σg ∅)))
76imbi2d 329 . . 3 (𝑥 = ∅ → ((𝑀 ∈ Mnd → (𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥))) ↔ (𝑀 ∈ Mnd → (𝑂 Σg ∅) = (𝑀 Σg ∅))))
8 oveq2 6557 . . . . 5 (𝑥 = 𝑦 → (𝑂 Σg 𝑥) = (𝑂 Σg 𝑦))
9 fveq2 6103 . . . . . 6 (𝑥 = 𝑦 → (reverse‘𝑥) = (reverse‘𝑦))
109oveq2d 6565 . . . . 5 (𝑥 = 𝑦 → (𝑀 Σg (reverse‘𝑥)) = (𝑀 Σg (reverse‘𝑦)))
118, 10eqeq12d 2625 . . . 4 (𝑥 = 𝑦 → ((𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥)) ↔ (𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦))))
1211imbi2d 329 . . 3 (𝑥 = 𝑦 → ((𝑀 ∈ Mnd → (𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥))) ↔ (𝑀 ∈ Mnd → (𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦)))))
13 oveq2 6557 . . . . 5 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝑂 Σg 𝑥) = (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)))
14 fveq2 6103 . . . . . 6 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (reverse‘𝑥) = (reverse‘(𝑦 ++ ⟨“𝑧”⟩)))
1514oveq2d 6565 . . . . 5 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → (𝑀 Σg (reverse‘𝑥)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))))
1613, 15eqeq12d 2625 . . . 4 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → ((𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥)) ↔ (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩)))))
1716imbi2d 329 . . 3 (𝑥 = (𝑦 ++ ⟨“𝑧”⟩) → ((𝑀 ∈ Mnd → (𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥))) ↔ (𝑀 ∈ Mnd → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))))))
18 oveq2 6557 . . . . 5 (𝑥 = 𝑊 → (𝑂 Σg 𝑥) = (𝑂 Σg 𝑊))
19 fveq2 6103 . . . . . 6 (𝑥 = 𝑊 → (reverse‘𝑥) = (reverse‘𝑊))
2019oveq2d 6565 . . . . 5 (𝑥 = 𝑊 → (𝑀 Σg (reverse‘𝑥)) = (𝑀 Σg (reverse‘𝑊)))
2118, 20eqeq12d 2625 . . . 4 (𝑥 = 𝑊 → ((𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥)) ↔ (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊))))
2221imbi2d 329 . . 3 (𝑥 = 𝑊 → ((𝑀 ∈ Mnd → (𝑂 Σg 𝑥) = (𝑀 Σg (reverse‘𝑥))) ↔ (𝑀 ∈ Mnd → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))))
23 gsumwrev.o . . . . . . 7 𝑂 = (oppg𝑀)
24 eqid 2610 . . . . . . 7 (0g𝑀) = (0g𝑀)
2523, 24oppgid 17609 . . . . . 6 (0g𝑀) = (0g𝑂)
2625gsum0 17101 . . . . 5 (𝑂 Σg ∅) = (0g𝑀)
2724gsum0 17101 . . . . 5 (𝑀 Σg ∅) = (0g𝑀)
2826, 27eqtr4i 2635 . . . 4 (𝑂 Σg ∅) = (𝑀 Σg ∅)
2928a1i 11 . . 3 (𝑀 ∈ Mnd → (𝑂 Σg ∅) = (𝑀 Σg ∅))
30 oveq2 6557 . . . . . 6 ((𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦)) → (𝑧(+g𝑀)(𝑂 Σg 𝑦)) = (𝑧(+g𝑀)(𝑀 Σg (reverse‘𝑦))))
3123oppgmnd 17607 . . . . . . . . . 10 (𝑀 ∈ Mnd → 𝑂 ∈ Mnd)
3231adantr 480 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → 𝑂 ∈ Mnd)
33 simprl 790 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → 𝑦 ∈ Word 𝐵)
34 simprr 792 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → 𝑧𝐵)
3534s1cld 13236 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ⟨“𝑧”⟩ ∈ Word 𝐵)
36 gsumwrev.b . . . . . . . . . . 11 𝐵 = (Base‘𝑀)
3723, 36oppgbas 17604 . . . . . . . . . 10 𝐵 = (Base‘𝑂)
38 eqid 2610 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
3937, 38gsumccat 17201 . . . . . . . . 9 ((𝑂 ∈ Mnd ∧ 𝑦 ∈ Word 𝐵 ∧ ⟨“𝑧”⟩ ∈ Word 𝐵) → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = ((𝑂 Σg 𝑦)(+g𝑂)(𝑂 Σg ⟨“𝑧”⟩)))
4032, 33, 35, 39syl3anc 1318 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = ((𝑂 Σg 𝑦)(+g𝑂)(𝑂 Σg ⟨“𝑧”⟩)))
4137gsumws1 17199 . . . . . . . . . . 11 (𝑧𝐵 → (𝑂 Σg ⟨“𝑧”⟩) = 𝑧)
4241ad2antll 761 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑂 Σg ⟨“𝑧”⟩) = 𝑧)
4342oveq2d 6565 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ((𝑂 Σg 𝑦)(+g𝑂)(𝑂 Σg ⟨“𝑧”⟩)) = ((𝑂 Σg 𝑦)(+g𝑂)𝑧))
44 eqid 2610 . . . . . . . . . 10 (+g𝑀) = (+g𝑀)
4544, 23, 38oppgplus 17602 . . . . . . . . 9 ((𝑂 Σg 𝑦)(+g𝑂)𝑧) = (𝑧(+g𝑀)(𝑂 Σg 𝑦))
4643, 45syl6eq 2660 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ((𝑂 Σg 𝑦)(+g𝑂)(𝑂 Σg ⟨“𝑧”⟩)) = (𝑧(+g𝑀)(𝑂 Σg 𝑦)))
4740, 46eqtrd 2644 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑧(+g𝑀)(𝑂 Σg 𝑦)))
48 revccat 13366 . . . . . . . . . . 11 ((𝑦 ∈ Word 𝐵 ∧ ⟨“𝑧”⟩ ∈ Word 𝐵) → (reverse‘(𝑦 ++ ⟨“𝑧”⟩)) = ((reverse‘⟨“𝑧”⟩) ++ (reverse‘𝑦)))
4933, 35, 48syl2anc 691 . . . . . . . . . 10 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (reverse‘(𝑦 ++ ⟨“𝑧”⟩)) = ((reverse‘⟨“𝑧”⟩) ++ (reverse‘𝑦)))
50 revs1 13365 . . . . . . . . . . 11 (reverse‘⟨“𝑧”⟩) = ⟨“𝑧”⟩
5150oveq1i 6559 . . . . . . . . . 10 ((reverse‘⟨“𝑧”⟩) ++ (reverse‘𝑦)) = (⟨“𝑧”⟩ ++ (reverse‘𝑦))
5249, 51syl6eq 2660 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (reverse‘(𝑦 ++ ⟨“𝑧”⟩)) = (⟨“𝑧”⟩ ++ (reverse‘𝑦)))
5352oveq2d 6565 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))) = (𝑀 Σg (⟨“𝑧”⟩ ++ (reverse‘𝑦))))
54 simpl 472 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → 𝑀 ∈ Mnd)
55 revcl 13361 . . . . . . . . . 10 (𝑦 ∈ Word 𝐵 → (reverse‘𝑦) ∈ Word 𝐵)
5655ad2antrl 760 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (reverse‘𝑦) ∈ Word 𝐵)
5736, 44gsumccat 17201 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ ⟨“𝑧”⟩ ∈ Word 𝐵 ∧ (reverse‘𝑦) ∈ Word 𝐵) → (𝑀 Σg (⟨“𝑧”⟩ ++ (reverse‘𝑦))) = ((𝑀 Σg ⟨“𝑧”⟩)(+g𝑀)(𝑀 Σg (reverse‘𝑦))))
5854, 35, 56, 57syl3anc 1318 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑀 Σg (⟨“𝑧”⟩ ++ (reverse‘𝑦))) = ((𝑀 Σg ⟨“𝑧”⟩)(+g𝑀)(𝑀 Σg (reverse‘𝑦))))
5936gsumws1 17199 . . . . . . . . . 10 (𝑧𝐵 → (𝑀 Σg ⟨“𝑧”⟩) = 𝑧)
6059ad2antll 761 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑀 Σg ⟨“𝑧”⟩) = 𝑧)
6160oveq1d 6564 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ((𝑀 Σg ⟨“𝑧”⟩)(+g𝑀)(𝑀 Σg (reverse‘𝑦))) = (𝑧(+g𝑀)(𝑀 Σg (reverse‘𝑦))))
6253, 58, 613eqtrd 2648 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))) = (𝑧(+g𝑀)(𝑀 Σg (reverse‘𝑦))))
6347, 62eqeq12d 2625 . . . . . 6 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ((𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))) ↔ (𝑧(+g𝑀)(𝑂 Σg 𝑦)) = (𝑧(+g𝑀)(𝑀 Σg (reverse‘𝑦)))))
6430, 63syl5ibr 235 . . . . 5 ((𝑀 ∈ Mnd ∧ (𝑦 ∈ Word 𝐵𝑧𝐵)) → ((𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦)) → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩)))))
6564expcom 450 . . . 4 ((𝑦 ∈ Word 𝐵𝑧𝐵) → (𝑀 ∈ Mnd → ((𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦)) → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))))))
6665a2d 29 . . 3 ((𝑦 ∈ Word 𝐵𝑧𝐵) → ((𝑀 ∈ Mnd → (𝑂 Σg 𝑦) = (𝑀 Σg (reverse‘𝑦))) → (𝑀 ∈ Mnd → (𝑂 Σg (𝑦 ++ ⟨“𝑧”⟩)) = (𝑀 Σg (reverse‘(𝑦 ++ ⟨“𝑧”⟩))))))
677, 12, 17, 22, 29, 66wrdind 13328 . 2 (𝑊 ∈ Word 𝐵 → (𝑀 ∈ Mnd → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊))))
6867impcom 445 1 ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149  reversecreverse 13152  Basecbs 15695  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  oppgcoppg 17598 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-reverse 13160  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-oppg 17599 This theorem is referenced by:  symgtrinv  17715
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