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Theorem mhmlem 12864
Description: Lemma for mhmmnd 12866 and ghmgrp 12868. (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 19 . 2 (𝜑𝜑)
2 mhmlem.a . 2 (𝜑𝐴𝑋)
3 mhmlem.b . 2 (𝜑𝐵𝑋)
4 eleq1 2240 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑋𝐴𝑋))
543anbi2d 1317 . . . . 5 (𝑥 = 𝐴 → ((𝜑𝑥𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝑦𝑋)))
6 fvoveq1 5892 . . . . . 6 (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝐴 + 𝑦)))
7 fveq2 5511 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
87oveq1d 5884 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝑦)))
96, 8eqeq12d 2192 . . . . 5 (𝑥 = 𝐴 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦))))
105, 9imbi12d 234 . . . 4 (𝑥 = 𝐴 → (((𝜑𝑥𝑋𝑦𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ ((𝜑𝐴𝑋𝑦𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)))))
11 eleq1 2240 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑋𝐵𝑋))
12113anbi3d 1318 . . . . 5 (𝑦 = 𝐵 → ((𝜑𝐴𝑋𝑦𝑋) ↔ (𝜑𝐴𝑋𝐵𝑋)))
13 oveq2 5877 . . . . . . 7 (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
1413fveq2d 5515 . . . . . 6 (𝑦 = 𝐵 → (𝐹‘(𝐴 + 𝑦)) = (𝐹‘(𝐴 + 𝐵)))
15 fveq2 5511 . . . . . . 7 (𝑦 = 𝐵 → (𝐹𝑦) = (𝐹𝐵))
1615oveq2d 5885 . . . . . 6 (𝑦 = 𝐵 → ((𝐹𝐴) (𝐹𝑦)) = ((𝐹𝐴) (𝐹𝐵)))
1714, 16eqeq12d 2192 . . . . 5 (𝑦 = 𝐵 → ((𝐹‘(𝐴 + 𝑦)) = ((𝐹𝐴) (𝐹𝑦)) ↔ (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
1812, 17imbi12d 234 . . . 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 2801 . . 3 ((𝐴𝑋𝐵𝑋) → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
212, 3, 20syl2anc 411 . 2 (𝜑 → ((𝜑𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵))))
221, 2, 3, 21mp3and 1340 1 (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹𝐴) (𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 978   = wceq 1353  wcel 2148  cfv 5212  (class class class)co 5869
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-iota 5174  df-fv 5220  df-ov 5872
This theorem is referenced by:  mhmid  12865  mhmmnd  12866  ghmgrp  12868
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