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Theorem lsmssass 32512
Description: Group sum is associative, subset version (see lsmass 19537). (Contributed by Thierry Arnoux, 1-Jun-2024.)
Hypotheses
Ref Expression
lsmssass.p = (LSSum‘𝐺)
lsmssass.b 𝐵 = (Base‘𝐺)
lsmssass.g (𝜑𝐺 ∈ Mnd)
lsmssass.r (𝜑𝑅𝐵)
lsmssass.t (𝜑𝑇𝐵)
lsmssass.u (𝜑𝑈𝐵)
Assertion
Ref Expression
lsmssass (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))

Proof of Theorem lsmssass
Dummy variables 𝑎 𝑐 𝑥 𝑦 𝑧 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmssass.g . . . . . . 7 (𝜑𝐺 ∈ Mnd)
2 lsmssass.r . . . . . . 7 (𝜑𝑅𝐵)
3 lsmssass.t . . . . . . 7 (𝜑𝑇𝐵)
4 lsmssass.b . . . . . . . 8 𝐵 = (Base‘𝐺)
5 eqid 2733 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmssass.p . . . . . . . 8 = (LSSum‘𝐺)
74, 5, 6lsmvalx 19507 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
81, 2, 3, 7syl3anc 1372 . . . . . 6 (𝜑 → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
98rexeqdv 3327 . . . . 5 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑦 ∈ ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
10 ovex 7442 . . . . . . 7 (𝑎(+g𝐺)𝑏) ∈ V
1110rgen2w 3067 . . . . . 6 𝑎𝑅𝑏𝑇 (𝑎(+g𝐺)𝑏) ∈ V
12 eqid 2733 . . . . . . 7 (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)) = (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))
13 oveq1 7416 . . . . . . . . 9 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑦(+g𝐺)𝑐) = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐))
1413eqeq2d 2744 . . . . . . . 8 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑥 = (𝑦(+g𝐺)𝑐) ↔ 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1514rexbidv 3179 . . . . . . 7 (𝑦 = (𝑎(+g𝐺)𝑏) → (∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1612, 15rexrnmpo 7548 . . . . . 6 (∀𝑎𝑅𝑏𝑇 (𝑎(+g𝐺)𝑏) ∈ V → (∃𝑦 ∈ ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1711, 16ax-mp 5 . . . . 5 (∃𝑦 ∈ ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐))
189, 17bitrdi 287 . . . 4 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
19 lsmssass.u . . . . . . . . . 10 (𝜑𝑈𝐵)
204, 5, 6lsmvalx 19507 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
211, 3, 19, 20syl3anc 1372 . . . . . . . . 9 (𝜑 → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
2221rexeqdv 3327 . . . . . . . 8 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑧 ∈ ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))𝑥 = (𝑎(+g𝐺)𝑧)))
23 ovex 7442 . . . . . . . . . 10 (𝑏(+g𝐺)𝑐) ∈ V
2423rgen2w 3067 . . . . . . . . 9 𝑏𝑇𝑐𝑈 (𝑏(+g𝐺)𝑐) ∈ V
25 eqid 2733 . . . . . . . . . 10 (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)) = (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))
26 oveq2 7417 . . . . . . . . . . 11 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑎(+g𝐺)𝑧) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
2726eqeq2d 2744 . . . . . . . . . 10 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑥 = (𝑎(+g𝐺)𝑧) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
2825, 27rexrnmpo 7548 . . . . . . . . 9 (∀𝑏𝑇𝑐𝑈 (𝑏(+g𝐺)𝑐) ∈ V → (∃𝑧 ∈ ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
2924, 28ax-mp 5 . . . . . . . 8 (∃𝑧 ∈ ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
3022, 29bitrdi 287 . . . . . . 7 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
3130adantr 482 . . . . . 6 ((𝜑𝑎𝑅) → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
321ad2antrr 725 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝐺 ∈ Mnd)
332ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑅𝐵)
34 simplr 768 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑎𝑅)
3533, 34sseldd 3984 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑎𝐵)
363ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑇𝐵)
37 simprl 770 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝑇)
3836, 37sseldd 3984 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝐵)
3919ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑈𝐵)
40 simprr 772 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝑈)
4139, 40sseldd 3984 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝐵)
424, 5mndass 18634 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4332, 35, 38, 41, 42syl13anc 1373 . . . . . . . 8 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4443eqeq2d 2744 . . . . . . 7 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → (𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
45442rexbidva 3218 . . . . . 6 ((𝜑𝑎𝑅) → (∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
4631, 45bitr4d 282 . . . . 5 ((𝜑𝑎𝑅) → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4746rexbidva 3177 . . . 4 (𝜑 → (∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4818, 47bitr4d 282 . . 3 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
494, 6lsmssv 19511 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) ⊆ 𝐵)
501, 2, 3, 49syl3anc 1372 . . . 4 (𝜑 → (𝑅 𝑇) ⊆ 𝐵)
514, 5, 6lsmelvalx 19508 . . . 4 ((𝐺 ∈ Mnd ∧ (𝑅 𝑇) ⊆ 𝐵𝑈𝐵) → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
521, 50, 19, 51syl3anc 1372 . . 3 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
534, 6lsmssv 19511 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)
541, 3, 19, 53syl3anc 1372 . . . 4 (𝜑 → (𝑇 𝑈) ⊆ 𝐵)
554, 5, 6lsmelvalx 19508 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑅𝐵 ∧ (𝑇 𝑈) ⊆ 𝐵) → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
561, 2, 54, 55syl3anc 1372 . . 3 (𝜑 → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
5748, 52, 563bitr4d 311 . 2 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ 𝑥 ∈ (𝑅 (𝑇 𝑈))))
5857eqrdv 2731 1 (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  wss 3949  ran crn 5678  cfv 6544  (class class class)co 7409  cmpo 7411  Basecbs 17144  +gcplusg 17197  Mndcmnd 18625  LSSumclsm 19502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-lsm 19504
This theorem is referenced by: (None)
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