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Theorem dfgrp3lem 17560
Description: Lemma for dfgrp3 17561. (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 1082 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐵 ≠ ∅)
2 n0 3964 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤𝐵)
31, 2sylib 208 . 2 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑤 𝑤𝐵)
4 oveq2 6698 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑙 + 𝑥) = (𝑙 + 𝑤))
54eqeq1d 2653 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑦))
65rexbidv 3081 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦))
7 oveq1 6697 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑥 + 𝑟) = (𝑤 + 𝑟))
87eqeq1d 2653 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑦))
98rexbidv 3081 . . . . . . . . 9 (𝑥 = 𝑤 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦))
106, 9anbi12d 747 . . . . . . . 8 (𝑥 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1110ralbidv 3015 . . . . . . 7 (𝑥 = 𝑤 → (∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
1211rspcv 3336 . . . . . 6 (𝑤𝐵 → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)))
13 eqeq2 2662 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑤))
1413rexbidv 3081 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤))
15 eqeq2 2662 . . . . . . . . . . 11 (𝑦 = 𝑤 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑤))
1615rexbidv 3081 . . . . . . . . . 10 (𝑦 = 𝑤 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
1714, 16anbi12d 747 . . . . . . . . 9 (𝑦 = 𝑤 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤)))
1817rspcva 3338 . . . . . . . 8 ((𝑤𝐵 ∧ ∀𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑤))
19 oveq1 6697 . . . . . . . . . . . 12 (𝑙 = 𝑢 → (𝑙 + 𝑤) = (𝑢 + 𝑤))
2019eqeq1d 2653 . . . . . . . . . . 11 (𝑙 = 𝑢 → ((𝑙 + 𝑤) = 𝑤 ↔ (𝑢 + 𝑤) = 𝑤))
2120cbvrexv 3202 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 ↔ ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2221biimpi 206 . . . . . . . . 9 (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑤 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
2322adantr 480 . . . . . . . 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 449 . . . . . 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 1104 . . . 4 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑤𝐵 → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤))
2827imp 444 . . 3 (((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵 (𝑢 + 𝑤) = 𝑤)
29 eqeq2 2662 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑙 + 𝑤) = 𝑦 ↔ (𝑙 + 𝑤) = 𝑎))
3029rexbidv 3081 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎))
31 eqeq2 2662 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑎 → ((𝑤 + 𝑟) = 𝑦 ↔ (𝑤 + 𝑟) = 𝑎))
3231rexbidv 3081 . . . . . . . . . . . . . . 15 (𝑦 = 𝑎 → (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3330, 32anbi12d 747 . . . . . . . . . . . . . 14 (𝑦 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑤) = 𝑦 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)))
3410, 33rspc2va 3354 . . . . . . . . . . . . 13 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑤) = 𝑎 ∧ ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3534simprd 478 . . . . . . . . . . . 12 (((𝑤𝐵𝑎𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3635expcom 450 . . . . . . . . . . 11 (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
37363ad2ant3 1104 . . . . . . . . . 10 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑤𝐵𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎))
3837impl 649 . . . . . . . . 9 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑎𝐵) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
3938ad2ant2r 798 . . . . . . . 8 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎)
40 oveq2 6698 . . . . . . . . . . . 12 (𝑟 = 𝑧 → (𝑤 + 𝑟) = (𝑤 + 𝑧))
4140eqeq1d 2653 . . . . . . . . . . 11 (𝑟 = 𝑧 → ((𝑤 + 𝑟) = 𝑎 ↔ (𝑤 + 𝑧) = 𝑎))
4241cbvrexv 3202 . . . . . . . . . 10 (∃𝑟𝐵 (𝑤 + 𝑟) = 𝑎 ↔ ∃𝑧𝐵 (𝑤 + 𝑧) = 𝑎)
43 simpll1 1120 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) → 𝐺 ∈ SGrp)
4443adantr 480 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝐺 ∈ SGrp)
45 simplr 807 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑢𝐵)
46 simpllr 815 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → 𝑤𝐵)
47 simprr 811 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 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 17337 . . . . . . . . . . . . . . 15 ((𝐺 ∈ SGrp ∧ (𝑢𝐵𝑤𝐵𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
5144, 45, 46, 47, 50syl13anc 1368 . . . . . . . . . . . . . 14 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑢 + (𝑤 + 𝑧)))
52 simprl 809 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + 𝑤) = 𝑤)
5352oveq1d 6705 . . . . . . . . . . . . . 14 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → ((𝑢 + 𝑤) + 𝑧) = (𝑤 + 𝑧))
5451, 53eqtr3d 2687 . . . . . . . . . . . . 13 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ ((𝑢 + 𝑤) = 𝑤𝑧𝐵)) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
5554anassrs 681 . . . . . . . . . . . 12 ((((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑢 + 𝑤) = 𝑤) ∧ 𝑧𝐵) → (𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧))
56 oveq2 6698 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑢 + (𝑤 + 𝑧)) = (𝑢 + 𝑎))
57 id 22 . . . . . . . . . . . . 13 ((𝑤 + 𝑧) = 𝑎 → (𝑤 + 𝑧) = 𝑎)
5856, 57eqeq12d 2666 . . . . . . . . . . . 12 ((𝑤 + 𝑧) = 𝑎 → ((𝑢 + (𝑤 + 𝑧)) = (𝑤 + 𝑧) ↔ (𝑢 + 𝑎) = 𝑎))
5955, 58syl5ibcom 235 . . . . . . . . . . 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))
6059rexlimdva 3060 . . . . . . . . . 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))
6142, 60syl5bi 232 . . . . . . . . 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))
6261adantrl 752 . . . . . . . 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))
6339, 62mpd 15 . . . . . . 7 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → (𝑢 + 𝑎) = 𝑎)
64 oveq2 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑙 + 𝑥) = (𝑙 + 𝑎))
6564eqeq1d 2653 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑙 + 𝑥) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑦))
6665rexbidv 3081 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦))
67 oveq1 6697 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → (𝑥 + 𝑟) = (𝑎 + 𝑟))
6867eqeq1d 2653 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑦))
6968rexbidv 3081 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → (∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦))
7066, 69anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → ((∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦)))
71 eqeq2 2662 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑙 + 𝑎) = 𝑦 ↔ (𝑙 + 𝑎) = 𝑢))
7271rexbidv 3081 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ↔ ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
73 eqeq2 2662 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑢 → ((𝑎 + 𝑟) = 𝑦 ↔ (𝑎 + 𝑟) = 𝑢))
7473rexbidv 3081 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦 ↔ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7572, 74anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((∃𝑙𝐵 (𝑙 + 𝑎) = 𝑦 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑦) ↔ (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢)))
7670, 75rspc2va 3354 . . . . . . . . . . . . . . . 16 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ∧ ∃𝑟𝐵 (𝑎 + 𝑟) = 𝑢))
7776simpld 474 . . . . . . . . . . . . . . 15 (((𝑎𝐵𝑢𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
7877ex 449 . . . . . . . . . . . . . 14 ((𝑎𝐵𝑢𝐵) → (∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
7978ancoms 468 . . . . . . . . . . . . 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 1104 . . . . . . . . . . 11 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑢𝐵𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢))
8281impl 649 . . . . . . . . . 10 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢)
83 oveq1 6697 . . . . . . . . . . . 12 (𝑙 = 𝑖 → (𝑙 + 𝑎) = (𝑖 + 𝑎))
8483eqeq1d 2653 . . . . . . . . . . 11 (𝑙 = 𝑖 → ((𝑙 + 𝑎) = 𝑢 ↔ (𝑖 + 𝑎) = 𝑢))
8584cbvrexv 3202 . . . . . . . . . 10 (∃𝑙𝐵 (𝑙 + 𝑎) = 𝑢 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8682, 85sylib 208 . . . . . . . . 9 ((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8786adantllr 755 . . . . . . . 8 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ 𝑎𝐵) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8887adantrr 753 . . . . . . 7 (((((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) ∧ 𝑢𝐵) ∧ (𝑎𝐵 ∧ (𝑢 + 𝑤) = 𝑤)) → ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)
8963, 88jca 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))
9089expr 642 . . . . 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)))
9190ralrimdva 2998 . . . 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)))
9291reximdva 3046 . . 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)))
9328, 92mpd 15 . 2 (((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) ∧ 𝑤𝐵) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
943, 93exlimddv 1903 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 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  c0 3948  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  SGrpcsgrp 17330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-sgrp 17331
This theorem is referenced by:  dfgrp3  17561
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