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Theorem dfgrp3mlem 13170
Description: Lemma for dfgrp3m 13171. (Contributed by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp3.b 𝐵 = (Base‘𝐺)
dfgrp3.p + = (+g𝐺)
Assertion
Ref Expression
dfgrp3mlem ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀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,𝑤,𝑥,𝑦   𝐺,𝑎,𝑖,𝑙,𝑟,𝑢,𝑤,𝑥,𝑦   + ,𝑎,𝑖,𝑙,𝑟,𝑢,𝑤,𝑥,𝑦

Proof of Theorem dfgrp3mlem
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1539 . . 3 𝑤 𝐺 ∈ Smgrp
2 nfe1 1507 . . 3 𝑤𝑤 𝑤𝐵
3 nfv 1539 . . 3 𝑤𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)
41, 2, 3nf3an 1577 . 2 𝑤(𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
5 nfv 1539 . 2 𝑤𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
6 simp2 1000 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑤 𝑤𝐵)
7 oveq2 5926 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑙 + 𝑥) = (𝑙 + 𝑤))
87eqeq1d 2202 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑦))
98rexbidv 2495 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦))
10 oveq1 5925 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥 + 𝑟) = (𝑤 + 𝑟))
1110eqeq1d 2202 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑦))
1211rexbidv 2495 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦))
139, 12anbi12d 473 . . . . . . . 8 (𝑥 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1413ralbidv 2494 . . . . . . 7 (𝑥 = 𝑤 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1514rspcv 2860 . . . . . 6 (𝑤𝐵 → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
16 eqeq2 2203 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑤))
1716rexbidv 2495 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤))
18 eqeq2 2203 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑤))
1918rexbidv 2495 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
2017, 19anbi12d 473 . . . . . . . . 9 (𝑦 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤)))
2120rspcva 2862 . . . . . . . 8 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
22 oveq1 5925 . . . . . . . . . . . 12 (𝑙 = 𝑢 → (𝑙 + 𝑤) = (𝑢 + 𝑤))
2322eqeq1d 2202 . . . . . . . . . . 11 (𝑙 = 𝑢 → ((𝑙 + 𝑤) = 𝑤 ↔ (𝑢 + 𝑤) = 𝑤))
2423cbvrexvw 2731 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ↔ ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2524biimpi 120 . . . . . . . . 9 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2625adantr 276 . . . . . . . 8 ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2721, 26syl 14 . . . . . . 7 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2827ex 115 . . . . . 6 (𝑤𝐵 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2915, 28syldc 46 . . . . 5 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
30293ad2ant3 1022 . . . 4 ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
3130imp 124 . . 3 (((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
32 eqeq2 2203 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑎))
3332rexbidv 2495 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎))
34 eqeq2 2203 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑎))
3534rexbidv 2495 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3633, 35anbi12d 473 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)))
3713, 36rspc2va 2878 . . . . . . . . . . . . 13 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3837simprd 114 . . . . . . . . . . . 12 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3938expcom 116 . . . . . . . . . . 11 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
40393ad2ant3 1022 . . . . . . . . . 10 ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
4140impl 380 . . . . . . . . 9 ((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
4241ad2ant2r 509 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
43 oveq2 5926 . . . . . . . . . . . 12 (𝑟 = 𝑧 → (𝑤 + 𝑟) = (𝑤 + 𝑧))
4443eqeq1d 2202 . . . . . . . . . . 11 (𝑟 = 𝑧 → ((𝑤 + 𝑟) = 𝑎 ↔ (𝑤 + 𝑧) = 𝑎))
4544cbvrexvw 2731 . . . . . . . . . 10 (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 ↔ ∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎)
46 simpll1 1038 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → 𝐺 ∈ Smgrp)
4746adantr 276 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝐺 ∈ Smgrp)
48 simplr 528 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑢𝐵)
49 simpllr 534 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑤𝐵)
50 simprr 531 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑧𝐵)
51 dfgrp3.b . . . . . . . . . . . . . . . 16 𝐵 = (Base‘𝐺)
52 dfgrp3.p . . . . . . . . . . . . . . . 16 + = (+g𝐺)
5351, 52sgrpass 12991 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Smgrp ∧ (𝑢𝐵𝑤𝐵𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
5447, 48, 49, 50, 53syl13anc 1251 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
55 simprl 529 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + 𝑤) = 𝑤)
5655oveq1d 5933 . . . . . . . . . . . . . 14 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑤 + 𝑧))
5754, 56eqtr3d 2228 . . . . . . . . . . . . 13 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
5857anassrs 400 . . . . . . . . . . . 12 ((((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
59 oveq2 5926 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑢 + (𝑤 + 𝑧)) = (𝑢 + 𝑎))
60 id 19 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑤 + 𝑧) = 𝑎)
6159, 60eqeq12d 2208 . . . . . . . . . . . 12 ((𝑤 + 𝑧) = 𝑎 → ((𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧) ↔ (𝑢 + 𝑎) = 𝑎))
6258, 61syl5ibcom 155 . . . . . . . . . . 11 ((((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → ((𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6362rexlimdva 2611 . . . . . . . . . 10 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6445, 63biimtrid 152 . . . . . . . . 9 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6564adantrl 478 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 → (𝑢 + 𝑎) = 𝑎))
6642, 65mpd 13 . . . . . . 7 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (𝑢 + 𝑎) = 𝑎)
67 oveq2 5926 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑙 + 𝑥) = (𝑙 + 𝑎))
6867eqeq1d 2202 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑦))
6968rexbidv 2495 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦))
70 oveq1 5925 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑥 + 𝑟) = (𝑎 + 𝑟))
7170eqeq1d 2202 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑦))
7271rexbidv 2495 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦))
7369, 72anbi12d 473 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦)))
74 eqeq2 2203 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑙 + 𝑎) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑢))
7574rexbidv 2495 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
76 eqeq2 2203 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑎 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑢))
7776rexbidv 2495 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7875, 77anbi12d 473 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢)))
7973, 78rspc2va 2878 . . . . . . . . . . . . . . . 16 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
8079simpld 112 . . . . . . . . . . . . . . 15 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
8180ex 115 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑢𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8281ancoms 268 . . . . . . . . . . . . 13 ((𝑢𝐵𝑎𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8382com12 30 . . . . . . . . . . . 12 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
84833ad2ant3 1022 . . . . . . . . . . 11 ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8584impl 380 . . . . . . . . . 10 ((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
86 oveq1 5925 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑙 + 𝑎) = (𝑖 + 𝑎))
8786eqeq1d 2202 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑙 + 𝑎) = 𝑢 ↔ (𝑖 + 𝑎) = 𝑢))
8887cbvrexvw 2731 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8985, 88sylib 122 . . . . . . . . 9 ((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
9089adantllr 481 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
9190adantrr 479 . . . . . . 7 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
9266, 91jca 306 . . . . . 6 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
9392expr 375 . . . . 5 (((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ((𝑢 + 𝑤) = 𝑤 → ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9493ralrimdva 2574 . . . 4 ((((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → ((𝑢 + 𝑤) = 𝑤 → ∀𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9594reximdva 2596 . . 3 (((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → (∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤 → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
9631, 95mpd 13 . 2 (((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
974, 5, 6, 96exlimdd 1883 1 ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wex 1503  wcel 2164  wral 2472  wrex 2473  cfv 5254  (class class class)co 5918  Basecbs 12618  +gcplusg 12695  Smgrpcsgrp 12984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-sgrp 12985
This theorem is referenced by:  dfgrp3m  13171
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