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Theorem dfgrp3lem 17336
 Description: Lemma for dfgrp3 17337. (Contributed by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp3.b 𝐵 = (Base‘𝐺)
dfgrp3.p + = (+g𝐺)
Assertion
Ref Expression
dfgrp3lem ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
Distinct variable groups:   𝐵,𝑎,𝑖,𝑙,𝑟,𝑢,𝑥,𝑦   𝐺,𝑎,𝑖,𝑙,𝑟,𝑢,𝑥,𝑦   + ,𝑎,𝑖,𝑙,𝑟,𝑢,𝑥,𝑦

Proof of Theorem dfgrp3lem
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1055 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐵 ≠ ∅)
2 n0 3890 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤𝐵)
31, 2sylib 207 . 2 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑤 𝑤𝐵)
4 oveq2 6557 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝑙 + 𝑥) = (𝑙 + 𝑤))
54eqeq1d 2612 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑦))
65rexbidv 3034 . . . . . . . . . 10 (𝑥 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦))
7 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = 𝑤 → (𝑥 + 𝑟) = (𝑤 + 𝑟))
87eqeq1d 2612 . . . . . . . . . . 11 (𝑥 = 𝑤 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑦))
98rexbidv 3034 . . . . . . . . . 10 (𝑥 = 𝑤 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦))
106, 9anbi12d 743 . . . . . . . . 9 (𝑥 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1110ralbidv 2969 . . . . . . . 8 (𝑥 = 𝑤 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1211rspcv 3278 . . . . . . 7 (𝑤𝐵 → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
13 eqeq2 2621 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑤))
1413rexbidv 3034 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤))
15 eqeq2 2621 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑤))
1615rexbidv 3034 . . . . . . . . . . 11 (𝑦 = 𝑤 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
1714, 16anbi12d 743 . . . . . . . . . 10 (𝑦 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤)))
1817rspcva 3280 . . . . . . . . 9 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
19 oveq1 6556 . . . . . . . . . . . . 13 (𝑙 = 𝑢 → (𝑙 + 𝑤) = (𝑢 + 𝑤))
2019eqeq1d 2612 . . . . . . . . . . . 12 (𝑙 = 𝑢 → ((𝑙 + 𝑤) = 𝑤 ↔ (𝑢 + 𝑤) = 𝑤))
2120cbvrexv 3148 . . . . . . . . . . 11 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ↔ ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2221biimpi 205 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2322adantr 480 . . . . . . . . 9 ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2418, 23syl 17 . . . . . . . 8 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2524ex 449 . . . . . . 7 (𝑤𝐵 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2612, 25syld 46 . . . . . 6 (𝑤𝐵 → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2726com12 32 . . . . 5 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
28273ad2ant3 1077 . . . 4 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2928imp 444 . . 3 (((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
30 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑎))
3130rexbidv 3034 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎))
32 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑎))
3332rexbidv 3034 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3431, 33anbi12d 743 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)))
3510, 34rspc2va 3294 . . . . . . . . . . . . 13 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3635simprd 478 . . . . . . . . . . . 12 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3736expcom 450 . . . . . . . . . . 11 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
38373ad2ant3 1077 . . . . . . . . . 10 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3938impl 648 . . . . . . . . 9 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
4039ad2ant2r 779 . . . . . . . 8 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
41 oveq2 6557 . . . . . . . . . . . 12 (𝑟 = 𝑧 → (𝑤 + 𝑟) = (𝑤 + 𝑧))
4241eqeq1d 2612 . . . . . . . . . . 11 (𝑟 = 𝑧 → ((𝑤 + 𝑟) = 𝑎 ↔ (𝑤 + 𝑧) = 𝑎))
4342cbvrexv 3148 . . . . . . . . . 10 (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 ↔ ∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎)
44 simpll1 1093 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → 𝐺 ∈ SGrp)
4544adantr 480 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝐺 ∈ SGrp)
46 simplr 788 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑢𝐵)
47 simpllr 795 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑤𝐵)
48 simprr 792 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑧𝐵)
49 dfgrp3.b . . . . . . . . . . . . . . . 16 𝐵 = (Base‘𝐺)
50 dfgrp3.p . . . . . . . . . . . . . . . 16 + = (+g𝐺)
5149, 50sgrpass 17113 . . . . . . . . . . . . . . 15 ((𝐺 ∈ SGrp ∧ (𝑢𝐵𝑤𝐵𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
5245, 46, 47, 48, 51syl13anc 1320 . . . . . . . . . . . . . 14 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
53 simprl 790 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + 𝑤) = 𝑤)
5453oveq1d 6564 . . . . . . . . . . . . . 14 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑤 + 𝑧))
5552, 54eqtr3d 2646 . . . . . . . . . . . . 13 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
5655anassrs 678 . . . . . . . . . . . 12 ((((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
57 oveq2 6557 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑢 + (𝑤 + 𝑧)) = (𝑢 + 𝑎))
58 id 22 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑤 + 𝑧) = 𝑎)
5957, 58eqeq12d 2625 . . . . . . . . . . . 12 ((𝑤 + 𝑧) = 𝑎 → ((𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧) ↔ (𝑢 + 𝑎) = 𝑎))
6056, 59syl5ibcom 234 . . . . . . . . . . 11 ((((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → ((𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6160rexlimdva 3013 . . . . . . . . . 10 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6243, 61syl5bi 231 . . . . . . . . 9 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6362adantrl 748 . . . . . . . 8 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6440, 63mpd 15 . . . . . . 7 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (𝑢 + 𝑎) = 𝑎)
65 oveq2 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑙 + 𝑥) = (𝑙 + 𝑎))
6665eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑦))
6766rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦))
68 oveq1 6556 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑥 + 𝑟) = (𝑎 + 𝑟))
6968eqeq1d 2612 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑦))
7069rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦))
7167, 70anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦)))
72 eqeq2 2621 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑙 + 𝑎) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑢))
7372rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
74 eqeq2 2621 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑎 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑢))
7574rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7673, 75anbi12d 743 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢)))
7771, 76rspc2va 3294 . . . . . . . . . . . . . . . 16 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7877simpld 474 . . . . . . . . . . . . . . 15 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
7978ex 449 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑢𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8079ancoms 468 . . . . . . . . . . . . 13 ((𝑢𝐵𝑎𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8180com12 32 . . . . . . . . . . . 12 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
82813ad2ant3 1077 . . . . . . . . . . 11 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8382impl 648 . . . . . . . . . 10 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
84 oveq1 6556 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑙 + 𝑎) = (𝑖 + 𝑎))
8584eqeq1d 2612 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑙 + 𝑎) = 𝑢 ↔ (𝑖 + 𝑎) = 𝑢))
8685cbvrexv 3148 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8783, 86sylib 207 . . . . . . . . 9 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8887adantllr 751 . . . . . . . 8 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8988adantrr 749 . . . . . . 7 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
9064, 89jca 553 . . . . . 6 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
9190expr 641 . . . . 5 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ((𝑢 + 𝑤) = 𝑤 → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9291ralrimdva 2952 . . . 4 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → ((𝑢 + 𝑤) = 𝑤 → ∀𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9392reximdva 3000 . . 3 (((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → (∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤 → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9429, 93mpd 15 . 2 (((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
953, 94exlimddv 1850 1 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  SGrpcsgrp 17106 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-sgrp 17107 This theorem is referenced by:  dfgrp3  17337
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