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Theorem symggrplem 18919
Description: Lemma for symggrp 19442 and efmndsgrp 18921. Conditions for an operation to be associative. Formerly part of proof for symggrp 19442. (Contributed by AV, 28-Jan-2024.)
Hypotheses
Ref Expression
symggrplem.c ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
symggrplem.p ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥𝑦))
Assertion
Ref Expression
symggrplem ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑦,𝑍   𝑥, + ,𝑦
Allowed substitution hint:   𝑍(𝑥)

Proof of Theorem symggrplem
StepHypRef Expression
1 coass 6296 . 2 ((𝑋𝑌) ∘ 𝑍) = (𝑋 ∘ (𝑌𝑍))
2 oveq1 7455 . . . . . 6 (𝑥 = 𝑋 → (𝑥 + 𝑦) = (𝑋 + 𝑦))
32eleq1d 2829 . . . . 5 (𝑥 = 𝑋 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑋 + 𝑦) ∈ 𝐵))
4 oveq2 7456 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
54eleq1d 2829 . . . . 5 (𝑦 = 𝑌 → ((𝑋 + 𝑦) ∈ 𝐵 ↔ (𝑋 + 𝑌) ∈ 𝐵))
6 symggrplem.c . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
73, 5, 6vtocl2ga 3590 . . . 4 ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
8 oveq1 7455 . . . . . 6 (𝑥 = (𝑋 + 𝑌) → (𝑥 + 𝑦) = ((𝑋 + 𝑌) + 𝑦))
9 coeq1 5882 . . . . . 6 (𝑥 = (𝑋 + 𝑌) → (𝑥𝑦) = ((𝑋 + 𝑌) ∘ 𝑦))
108, 9eqeq12d 2756 . . . . 5 (𝑥 = (𝑋 + 𝑌) → ((𝑥 + 𝑦) = (𝑥𝑦) ↔ ((𝑋 + 𝑌) + 𝑦) = ((𝑋 + 𝑌) ∘ 𝑦)))
11 oveq2 7456 . . . . . 6 (𝑦 = 𝑍 → ((𝑋 + 𝑌) + 𝑦) = ((𝑋 + 𝑌) + 𝑍))
12 coeq2 5883 . . . . . 6 (𝑦 = 𝑍 → ((𝑋 + 𝑌) ∘ 𝑦) = ((𝑋 + 𝑌) ∘ 𝑍))
1311, 12eqeq12d 2756 . . . . 5 (𝑦 = 𝑍 → (((𝑋 + 𝑌) + 𝑦) = ((𝑋 + 𝑌) ∘ 𝑦) ↔ ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑌) ∘ 𝑍)))
14 symggrplem.p . . . . 5 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑥𝑦))
1510, 13, 14vtocl2ga 3590 . . . 4 (((𝑋 + 𝑌) ∈ 𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑌) ∘ 𝑍))
167, 15stoic3 1774 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑌) ∘ 𝑍))
17 coeq1 5882 . . . . . . 7 (𝑥 = 𝑋 → (𝑥𝑦) = (𝑋𝑦))
182, 17eqeq12d 2756 . . . . . 6 (𝑥 = 𝑋 → ((𝑥 + 𝑦) = (𝑥𝑦) ↔ (𝑋 + 𝑦) = (𝑋𝑦)))
19 coeq2 5883 . . . . . . 7 (𝑦 = 𝑌 → (𝑋𝑦) = (𝑋𝑌))
204, 19eqeq12d 2756 . . . . . 6 (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑋𝑦) ↔ (𝑋 + 𝑌) = (𝑋𝑌)))
2118, 20, 14vtocl2ga 3590 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))
22213adant3 1132 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))
2322coeq1d 5886 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) ∘ 𝑍) = ((𝑋𝑌) ∘ 𝑍))
2416, 23eqtrd 2780 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋𝑌) ∘ 𝑍))
25 simp1 1136 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → 𝑋𝐵)
26 oveq1 7455 . . . . . . 7 (𝑥 = 𝑌 → (𝑥 + 𝑦) = (𝑌 + 𝑦))
2726eleq1d 2829 . . . . . 6 (𝑥 = 𝑌 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑌 + 𝑦) ∈ 𝐵))
28 oveq2 7456 . . . . . . 7 (𝑦 = 𝑍 → (𝑌 + 𝑦) = (𝑌 + 𝑍))
2928eleq1d 2829 . . . . . 6 (𝑦 = 𝑍 → ((𝑌 + 𝑦) ∈ 𝐵 ↔ (𝑌 + 𝑍) ∈ 𝐵))
3027, 29, 6vtocl2ga 3590 . . . . 5 ((𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
31303adant1 1130 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) ∈ 𝐵)
32 oveq2 7456 . . . . . 6 (𝑦 = (𝑌 + 𝑍) → (𝑋 + 𝑦) = (𝑋 + (𝑌 + 𝑍)))
33 coeq2 5883 . . . . . 6 (𝑦 = (𝑌 + 𝑍) → (𝑋𝑦) = (𝑋 ∘ (𝑌 + 𝑍)))
3432, 33eqeq12d 2756 . . . . 5 (𝑦 = (𝑌 + 𝑍) → ((𝑋 + 𝑦) = (𝑋𝑦) ↔ (𝑋 + (𝑌 + 𝑍)) = (𝑋 ∘ (𝑌 + 𝑍))))
3518, 34, 14vtocl2ga 3590 . . . 4 ((𝑋𝐵 ∧ (𝑌 + 𝑍) ∈ 𝐵) → (𝑋 + (𝑌 + 𝑍)) = (𝑋 ∘ (𝑌 + 𝑍)))
3625, 31, 35syl2anc 583 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 + (𝑌 + 𝑍)) = (𝑋 ∘ (𝑌 + 𝑍)))
37 coeq1 5882 . . . . . . 7 (𝑥 = 𝑌 → (𝑥𝑦) = (𝑌𝑦))
3826, 37eqeq12d 2756 . . . . . 6 (𝑥 = 𝑌 → ((𝑥 + 𝑦) = (𝑥𝑦) ↔ (𝑌 + 𝑦) = (𝑌𝑦)))
39 coeq2 5883 . . . . . . 7 (𝑦 = 𝑍 → (𝑌𝑦) = (𝑌𝑍))
4028, 39eqeq12d 2756 . . . . . 6 (𝑦 = 𝑍 → ((𝑌 + 𝑦) = (𝑌𝑦) ↔ (𝑌 + 𝑍) = (𝑌𝑍)))
4138, 40, 14vtocl2ga 3590 . . . . 5 ((𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑌𝑍))
42413adant1 1130 . . . 4 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑌 + 𝑍) = (𝑌𝑍))
4342coeq2d 5887 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 ∘ (𝑌 + 𝑍)) = (𝑋 ∘ (𝑌𝑍)))
4436, 43eqtrd 2780 . 2 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (𝑋 + (𝑌 + 𝑍)) = (𝑋 ∘ (𝑌𝑍)))
451, 24, 443eqtr4a 2806 1 ((𝑋𝐵𝑌𝐵𝑍𝐵) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1537  wcel 2108  ccom 5704  (class class class)co 7448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-co 5709  df-iota 6525  df-fv 6581  df-ov 7451
This theorem is referenced by:  efmndsgrp  18921  smndex1sgrp  18943  symggrp  19442
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