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Theorem lsmssass 31013
Description: Group sum is associative, subset version (see lsmass 18791). (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 2801 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmssass.p . . . . . . . 8 = (LSSum‘𝐺)
74, 5, 6lsmvalx 18760 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
81, 2, 3, 7syl3anc 1368 . . . . . 6 (𝜑 → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
98rexeqdv 3368 . . . . 5 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑦 ∈ ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
10 ovex 7172 . . . . . . 7 (𝑎(+g𝐺)𝑏) ∈ V
1110rgen2w 3122 . . . . . 6 𝑎𝑅𝑏𝑇 (𝑎(+g𝐺)𝑏) ∈ V
12 eqid 2801 . . . . . . 7 (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)) = (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))
13 oveq1 7146 . . . . . . . . 9 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑦(+g𝐺)𝑐) = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐))
1413eqeq2d 2812 . . . . . . . 8 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑥 = (𝑦(+g𝐺)𝑐) ↔ 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1514rexbidv 3259 . . . . . . 7 (𝑦 = (𝑎(+g𝐺)𝑏) → (∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1612, 15rexrnmpo 7273 . . . . . 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, 17syl6bb 290 . . . 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 18760 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
211, 3, 19, 20syl3anc 1368 . . . . . . . . 9 (𝜑 → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
2221rexeqdv 3368 . . . . . . . 8 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑧 ∈ ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))𝑥 = (𝑎(+g𝐺)𝑧)))
23 ovex 7172 . . . . . . . . . 10 (𝑏(+g𝐺)𝑐) ∈ V
2423rgen2w 3122 . . . . . . . . 9 𝑏𝑇𝑐𝑈 (𝑏(+g𝐺)𝑐) ∈ V
25 eqid 2801 . . . . . . . . . 10 (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)) = (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))
26 oveq2 7147 . . . . . . . . . . 11 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑎(+g𝐺)𝑧) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
2726eqeq2d 2812 . . . . . . . . . 10 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑥 = (𝑎(+g𝐺)𝑧) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
2825, 27rexrnmpo 7273 . . . . . . . . 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, 29syl6bb 290 . . . . . . 7 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
3130adantr 484 . . . . . 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 3919 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑎𝐵)
363ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑇𝐵)
37 simprl 770 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝑇)
3836, 37sseldd 3919 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝐵)
3919ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑈𝐵)
40 simprr 772 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝑈)
4139, 40sseldd 3919 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝐵)
424, 5mndass 17916 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4332, 35, 38, 41, 42syl13anc 1369 . . . . . . . 8 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4443eqeq2d 2812 . . . . . . 7 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → (𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
45442rexbidva 3261 . . . . . 6 ((𝜑𝑎𝑅) → (∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
4631, 45bitr4d 285 . . . . 5 ((𝜑𝑎𝑅) → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4746rexbidva 3258 . . . 4 (𝜑 → (∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4818, 47bitr4d 285 . . 3 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
494, 6lsmssv 18764 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) ⊆ 𝐵)
501, 2, 3, 49syl3anc 1368 . . . 4 (𝜑 → (𝑅 𝑇) ⊆ 𝐵)
514, 5, 6lsmelvalx 18761 . . . 4 ((𝐺 ∈ Mnd ∧ (𝑅 𝑇) ⊆ 𝐵𝑈𝐵) → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
521, 50, 19, 51syl3anc 1368 . . 3 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
534, 6lsmssv 18764 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)
541, 3, 19, 53syl3anc 1368 . . . 4 (𝜑 → (𝑇 𝑈) ⊆ 𝐵)
554, 5, 6lsmelvalx 18761 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑅𝐵 ∧ (𝑇 𝑈) ⊆ 𝐵) → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
561, 2, 54, 55syl3anc 1368 . . 3 (𝜑 → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
5748, 52, 563bitr4d 314 . 2 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ 𝑥 ∈ (𝑅 (𝑇 𝑈))))
5857eqrdv 2799 1 (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  wss 3884  ran crn 5524  cfv 6328  (class class class)co 7139  cmpo 7141  Basecbs 16479  +gcplusg 16561  Mndcmnd 17907  LSSumclsm 18755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-lsm 18757
This theorem is referenced by: (None)
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