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Theorem mhmlem 17358
 Description: Lemma for mhmmnd 17360 and ghmgrp 17362. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)
Hypotheses
Ref Expression
ghmgrp.f ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
mhmlem.a (𝜑𝐴𝑋)
mhmlem.b (𝜑𝐵𝑋)
Assertion
Ref Expression
mhmlem (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦   𝑥, ,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mhmlem
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 mhmlem.a . 2 (𝜑𝐴𝑋)
3 mhmlem.b . 2 (𝜑𝐵𝑋)
4 eleq1 2676 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
543anbi2d 1396 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝑦𝑋)))
6 oveq1 6556 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
76fveq2d 6107 . . . . . 6 (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝐴 + 𝑦)))
8 fveq2 6103 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
98oveq1d 6564 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝑦)))
107, 9eqeq12d 2625 . . . . 5 (𝑥 = 𝐴 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦))))
115, 10imbi12d 333 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ ((𝜑𝐴𝑋𝑦𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)))))
12 eleq1 2676 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
13123anbi3d 1397 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝐵𝑋)))
14 oveq2 6557 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1514fveq2d 6107 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘(𝐴 + 𝑦)) = (𝐹‘(𝐴 + 𝐵)))
16 fveq2 6103 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1716oveq2d 6565 . . . . . 6 (𝑦 = 𝐵 → ((𝐹𝐴) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝐵)))
1815, 17eqeq12d 2625 . . . . 5 (𝑦 = 𝐵 → ((𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
1913, 18imbi12d 333 . . . 4 (𝑦 = 𝐵 → (((𝜑𝐴𝑋𝑦𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦))) ↔ ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))))
20 ghmgrp.f . . . 4 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
2111, 19, 20vtocl2g 3243 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
222, 3, 21syl2anc 691 . 2 (𝜑 → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
231, 2, 3, 22mp3and 1419 1 (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549 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 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-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  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 This theorem is referenced by:  mhmid  17359  mhmmnd  17360  ghmgrp  17362  ghmcmn  18060
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