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Theorem mhmlem 18151
Description: Lemma for mhmmnd 18153 and ghmgrp 18155. (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 2904 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
543anbi2d 1434 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝑦𝑋)))
6 fvoveq1 7174 . . . . . 6 (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝐴 + 𝑦)))
7 fveq2 6666 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
87oveq1d 7166 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝑦)))
96, 8eqeq12d 2840 . . . . 5 (𝑥 = 𝐴 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦))))
105, 9imbi12d 346 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ ((𝜑𝐴𝑋𝑦𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)))))
11 eleq1 2904 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
12113anbi3d 1435 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝐵𝑋)))
13 oveq2 7159 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1413fveq2d 6670 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘(𝐴 + 𝑦)) = (𝐹‘(𝐴 + 𝐵)))
15 fveq2 6666 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1615oveq2d 7167 . . . . . 6 (𝑦 = 𝐵 → ((𝐹𝐴) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝐵)))
1714, 16eqeq12d 2840 . . . . 5 (𝑦 = 𝐵 → ((𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
1812, 17imbi12d 346 . . . 4 (𝑦 = 𝐵 → (((𝜑𝐴𝑋𝑦𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦))) ↔ ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))))
19 ghmgrp.f . . . 4 ((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
2010, 18, 19vtocl2g 3576 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
212, 3, 20syl2anc 584 . 2 (𝜑 → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
221, 2, 3, 21mp3and 1457 1 (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2106  cfv 6351  (class class class)co 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-ov 7154
This theorem is referenced by:  mhmid  18152  mhmmnd  18153  ghmgrp  18155  ghmcmn  18874
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