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Theorem dfgrp3lem 18264
 Description: Lemma for dfgrp3 18265. (Contributed by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp3.b 𝐵 = (Base‘𝐺)
dfgrp3.p + = (+g𝐺)
Assertion
Ref Expression
dfgrp3lem ((𝐺 ∈ Smgrp ∧ 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 1134 . . 3 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐵 ≠ ∅)
2 n0 4245 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤𝐵)
31, 2sylib 221 . 2 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑤 𝑤𝐵)
4 oveq2 7158 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑙 + 𝑥) = (𝑙 + 𝑤))
54eqeq1d 2760 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑦))
65rexbidv 3221 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦))
7 oveq1 7157 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥 + 𝑟) = (𝑤 + 𝑟))
87eqeq1d 2760 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑦))
98rexbidv 3221 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦))
106, 9anbi12d 633 . . . . . . . 8 (𝑥 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1110ralbidv 3126 . . . . . . 7 (𝑥 = 𝑤 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1211rspcv 3536 . . . . . 6 (𝑤𝐵 → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
13 eqeq2 2770 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑤))
1413rexbidv 3221 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤))
15 eqeq2 2770 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑤))
1615rexbidv 3221 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
1714, 16anbi12d 633 . . . . . . . . 9 (𝑦 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤)))
1817rspcva 3539 . . . . . . . 8 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
19 oveq1 7157 . . . . . . . . . . . 12 (𝑙 = 𝑢 → (𝑙 + 𝑤) = (𝑢 + 𝑤))
2019eqeq1d 2760 . . . . . . . . . . 11 (𝑙 = 𝑢 → ((𝑙 + 𝑤) = 𝑤 ↔ (𝑢 + 𝑤) = 𝑤))
2120cbvrexvw 3362 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ↔ ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2221biimpi 219 . . . . . . . . 9 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2322adantr 484 . . . . . . . 8 ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2418, 23syl 17 . . . . . . 7 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2524ex 416 . . . . . 6 (𝑤𝐵 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2612, 25syldc 48 . . . . 5 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
27263ad2ant3 1132 . . . 4 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2827imp 410 . . 3 (((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
29 eqeq2 2770 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑎))
3029rexbidv 3221 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎))
31 eqeq2 2770 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑎))
3231rexbidv 3221 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3330, 32anbi12d 633 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)))
3410, 33rspc2va 3552 . . . . . . . . . . . . 13 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3534simprd 499 . . . . . . . . . . . 12 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3635expcom 417 . . . . . . . . . . 11 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
37363ad2ant3 1132 . . . . . . . . . 10 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3837impl 459 . . . . . . . . 9 ((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3938ad2ant2r 746 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
40 oveq2 7158 . . . . . . . . . . . 12 (𝑟 = 𝑧 → (𝑤 + 𝑟) = (𝑤 + 𝑧))
4140eqeq1d 2760 . . . . . . . . . . 11 (𝑟 = 𝑧 → ((𝑤 + 𝑟) = 𝑎 ↔ (𝑤 + 𝑧) = 𝑎))
4241cbvrexvw 3362 . . . . . . . . . 10 (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 ↔ ∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎)
43 simpll1 1209 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → 𝐺 ∈ Smgrp)
4443adantr 484 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝐺 ∈ Smgrp)
45 simplr 768 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑢𝐵)
46 simpllr 775 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑤𝐵)
47 simprr 772 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑧𝐵)
48 dfgrp3.b . . . . . . . . . . . . . . . 16 𝐵 = (Base‘𝐺)
49 dfgrp3.p . . . . . . . . . . . . . . . 16 + = (+g𝐺)
5048, 49sgrpass 17973 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Smgrp ∧ (𝑢𝐵𝑤𝐵𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
5144, 45, 46, 47, 50syl13anc 1369 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
52 simprl 770 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + 𝑤) = 𝑤)
5352oveq1d 7165 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑤 + 𝑧))
5451, 53eqtr3d 2795 . . . . . . . . . . . . 13 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
5554anassrs 471 . . . . . . . . . . . 12 ((((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
56 oveq2 7158 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑢 + (𝑤 + 𝑧)) = (𝑢 + 𝑎))
57 id 22 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑤 + 𝑧) = 𝑎)
5856, 57eqeq12d 2774 . . . . . . . . . . . 12 ((𝑤 + 𝑧) = 𝑎 → ((𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧) ↔ (𝑢 + 𝑎) = 𝑎))
5955, 58syl5ibcom 248 . . . . . . . . . . 11 ((((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → ((𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6059rexlimdva 3208 . . . . . . . . . 10 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6142, 60syl5bi 245 . . . . . . . . 9 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6261adantrl 715 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6339, 62mpd 15 . . . . . . 7 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (𝑢 + 𝑎) = 𝑎)
64 oveq2 7158 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑙 + 𝑥) = (𝑙 + 𝑎))
6564eqeq1d 2760 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑦))
6665rexbidv 3221 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦))
67 oveq1 7157 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑥 + 𝑟) = (𝑎 + 𝑟))
6867eqeq1d 2760 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑦))
6968rexbidv 3221 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦))
7066, 69anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦)))
71 eqeq2 2770 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑙 + 𝑎) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑢))
7271rexbidv 3221 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
73 eqeq2 2770 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑎 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑢))
7473rexbidv 3221 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7572, 74anbi12d 633 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢)))
7670, 75rspc2va 3552 . . . . . . . . . . . . . . . 16 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7776simpld 498 . . . . . . . . . . . . . . 15 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
7877ex 416 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑢𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
7978ancoms 462 . . . . . . . . . . . . 13 ((𝑢𝐵𝑎𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8079com12 32 . . . . . . . . . . . 12 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
81803ad2ant3 1132 . . . . . . . . . . 11 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8281impl 459 . . . . . . . . . 10 ((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
83 oveq1 7157 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑙 + 𝑎) = (𝑖 + 𝑎))
8483eqeq1d 2760 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑙 + 𝑎) = 𝑢 ↔ (𝑖 + 𝑎) = 𝑢))
8584cbvrexvw 3362 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8682, 85sylib 221 . . . . . . . . 9 ((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8786adantllr 718 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8887adantrr 716 . . . . . . 7 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8963, 88jca 515 . . . . . 6 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
9089expr 460 . . . . 5 (((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ((𝑢 + 𝑤) = 𝑤 → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9190ralrimdva 3118 . . . 4 ((((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → ((𝑢 + 𝑤) = 𝑤 → ∀𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9291reximdva 3198 . . 3 (((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → (∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤 → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9328, 92mpd 15 . 2 (((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
943, 93exlimddv 1936 1 ((𝐺 ∈ Smgrp ∧ 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 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  ∅c0 4225  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  Smgrpcsgrp 17966 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-ov 7153  df-sgrp 17967 This theorem is referenced by:  dfgrp3  18265
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