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Theorem lsmssass 32183
Description: Group sum is associative, subset version (see lsmass 19451). (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 2736 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmssass.p . . . . . . . 8 = (LSSum‘𝐺)
74, 5, 6lsmvalx 19421 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
81, 2, 3, 7syl3anc 1371 . . . . . 6 (𝜑 → (𝑅 𝑇) = ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)))
98rexeqdv 3314 . . . . 5 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑦 ∈ ran (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
10 ovex 7390 . . . . . . 7 (𝑎(+g𝐺)𝑏) ∈ V
1110rgen2w 3069 . . . . . 6 𝑎𝑅𝑏𝑇 (𝑎(+g𝐺)𝑏) ∈ V
12 eqid 2736 . . . . . . 7 (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏)) = (𝑎𝑅, 𝑏𝑇 ↦ (𝑎(+g𝐺)𝑏))
13 oveq1 7364 . . . . . . . . 9 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑦(+g𝐺)𝑐) = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐))
1413eqeq2d 2747 . . . . . . . 8 (𝑦 = (𝑎(+g𝐺)𝑏) → (𝑥 = (𝑦(+g𝐺)𝑐) ↔ 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1514rexbidv 3175 . . . . . . 7 (𝑦 = (𝑎(+g𝐺)𝑏) → (∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
1612, 15rexrnmpo 7495 . . . . . 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 286 . . . 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 19421 . . . . . . . . . 10 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
211, 3, 19, 20syl3anc 1371 . . . . . . . . 9 (𝜑 → (𝑇 𝑈) = ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)))
2221rexeqdv 3314 . . . . . . . 8 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑧 ∈ ran (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))𝑥 = (𝑎(+g𝐺)𝑧)))
23 ovex 7390 . . . . . . . . . 10 (𝑏(+g𝐺)𝑐) ∈ V
2423rgen2w 3069 . . . . . . . . 9 𝑏𝑇𝑐𝑈 (𝑏(+g𝐺)𝑐) ∈ V
25 eqid 2736 . . . . . . . . . 10 (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐)) = (𝑏𝑇, 𝑐𝑈 ↦ (𝑏(+g𝐺)𝑐))
26 oveq2 7365 . . . . . . . . . . 11 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑎(+g𝐺)𝑧) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
2726eqeq2d 2747 . . . . . . . . . 10 (𝑧 = (𝑏(+g𝐺)𝑐) → (𝑥 = (𝑎(+g𝐺)𝑧) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
2825, 27rexrnmpo 7495 . . . . . . . . 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 286 . . . . . . 7 (𝜑 → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
3130adantr 481 . . . . . 6 ((𝜑𝑎𝑅) → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
321ad2antrr 724 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝐺 ∈ Mnd)
332ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑅𝐵)
34 simplr 767 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑎𝑅)
3533, 34sseldd 3945 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑎𝐵)
363ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑇𝐵)
37 simprl 769 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝑇)
3836, 37sseldd 3945 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑏𝐵)
3919ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑈𝐵)
40 simprr 771 . . . . . . . . . 10 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝑈)
4139, 40sseldd 3945 . . . . . . . . 9 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → 𝑐𝐵)
424, 5mndass 18565 . . . . . . . . 9 ((𝐺 ∈ Mnd ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4332, 35, 38, 41, 42syl13anc 1372 . . . . . . . 8 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐)))
4443eqeq2d 2747 . . . . . . 7 (((𝜑𝑎𝑅) ∧ (𝑏𝑇𝑐𝑈)) → (𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
45442rexbidva 3211 . . . . . 6 ((𝜑𝑎𝑅) → (∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)(𝑏(+g𝐺)𝑐))))
4631, 45bitr4d 281 . . . . 5 ((𝜑𝑎𝑅) → (∃𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4746rexbidva 3173 . . . 4 (𝜑 → (∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧) ↔ ∃𝑎𝑅𝑏𝑇𝑐𝑈 𝑥 = ((𝑎(+g𝐺)𝑏)(+g𝐺)𝑐)))
4818, 47bitr4d 281 . . 3 (𝜑 → (∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
494, 6lsmssv 19425 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑅𝐵𝑇𝐵) → (𝑅 𝑇) ⊆ 𝐵)
501, 2, 3, 49syl3anc 1371 . . . 4 (𝜑 → (𝑅 𝑇) ⊆ 𝐵)
514, 5, 6lsmelvalx 19422 . . . 4 ((𝐺 ∈ Mnd ∧ (𝑅 𝑇) ⊆ 𝐵𝑈𝐵) → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
521, 50, 19, 51syl3anc 1371 . . 3 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ ∃𝑦 ∈ (𝑅 𝑇)∃𝑐𝑈 𝑥 = (𝑦(+g𝐺)𝑐)))
534, 6lsmssv 19425 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑇𝐵𝑈𝐵) → (𝑇 𝑈) ⊆ 𝐵)
541, 3, 19, 53syl3anc 1371 . . . 4 (𝜑 → (𝑇 𝑈) ⊆ 𝐵)
554, 5, 6lsmelvalx 19422 . . . 4 ((𝐺 ∈ Mnd ∧ 𝑅𝐵 ∧ (𝑇 𝑈) ⊆ 𝐵) → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
561, 2, 54, 55syl3anc 1371 . . 3 (𝜑 → (𝑥 ∈ (𝑅 (𝑇 𝑈)) ↔ ∃𝑎𝑅𝑧 ∈ (𝑇 𝑈)𝑥 = (𝑎(+g𝐺)𝑧)))
5748, 52, 563bitr4d 310 . 2 (𝜑 → (𝑥 ∈ ((𝑅 𝑇) 𝑈) ↔ 𝑥 ∈ (𝑅 (𝑇 𝑈))))
5857eqrdv 2734 1 (𝜑 → ((𝑅 𝑇) 𝑈) = (𝑅 (𝑇 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133  Mndcmnd 18556  LSSumclsm 19416
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-lsm 19418
This theorem is referenced by: (None)
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