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Theorem islmhm2 18952
Description: A one-equation proof of linearity of a left module homomorphism, similar to df-lss 18847. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
islmhm2.b 𝐵 = (Base‘𝑆)
islmhm2.c 𝐶 = (Base‘𝑇)
islmhm2.k 𝐾 = (Scalar‘𝑆)
islmhm2.l 𝐿 = (Scalar‘𝑇)
islmhm2.e 𝐸 = (Base‘𝐾)
islmhm2.p + = (+g𝑆)
islmhm2.q = (+g𝑇)
islmhm2.m · = ( ·𝑠𝑆)
islmhm2.n × = ( ·𝑠𝑇)
Assertion
Ref Expression
islmhm2 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))))
Distinct variable groups:   𝑥,𝑦,𝑧,   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐸,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧   𝑥, · ,𝑧   𝑥, × ,𝑧
Allowed substitution hints:   · (𝑦)   × (𝑦)

Proof of Theorem islmhm2
StepHypRef Expression
1 islmhm2.b . . . . 5 𝐵 = (Base‘𝑆)
2 islmhm2.c . . . . 5 𝐶 = (Base‘𝑇)
31, 2lmhmf 18948 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵𝐶)
4 islmhm2.k . . . . 5 𝐾 = (Scalar‘𝑆)
5 islmhm2.l . . . . 5 𝐿 = (Scalar‘𝑇)
64, 5lmhmsca 18944 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)
7 lmghm 18945 . . . . . . . 8 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
87adantr 481 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
9 lmhmlmod1 18947 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109adantr 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → 𝑆 ∈ LMod)
11 simpr1 1065 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → 𝑥𝐸)
12 simpr2 1066 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
13 islmhm2.m . . . . . . . . 9 · = ( ·𝑠𝑆)
14 islmhm2.e . . . . . . . . 9 𝐸 = (Base‘𝐾)
151, 4, 13, 14lmodvscl 18796 . . . . . . . 8 ((𝑆 ∈ LMod ∧ 𝑥𝐸𝑦𝐵) → (𝑥 · 𝑦) ∈ 𝐵)
1610, 11, 12, 15syl3anc 1323 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
17 simpr3 1067 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → 𝑧𝐵)
18 islmhm2.p . . . . . . . 8 + = (+g𝑆)
19 islmhm2.q . . . . . . . 8 = (+g𝑇)
201, 18, 19ghmlin 17581 . . . . . . 7 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑥 · 𝑦) ∈ 𝐵𝑧𝐵) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝐹‘(𝑥 · 𝑦)) (𝐹𝑧)))
218, 16, 17, 20syl3anc 1323 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝐹‘(𝑥 · 𝑦)) (𝐹𝑧)))
22 islmhm2.n . . . . . . . . 9 × = ( ·𝑠𝑇)
234, 14, 1, 13, 22lmhmlin 18949 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑥𝐸𝑦𝐵) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
24233adant3r3 1273 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
2524oveq1d 6620 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → ((𝐹‘(𝑥 · 𝑦)) (𝐹𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))
2621, 25eqtrd 2660 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ (𝑥𝐸𝑦𝐵𝑧𝐵)) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))
2726ralrimivvva 2971 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))
283, 6, 273jca 1240 . . 3 (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧))))
2928adantl 482 . 2 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧))))
30 lmodgrp 18786 . . . . . 6 (𝑆 ∈ LMod → 𝑆 ∈ Grp)
31 lmodgrp 18786 . . . . . 6 (𝑇 ∈ LMod → 𝑇 ∈ Grp)
3230, 31anim12i 589 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3332adantr 481 . . . 4 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
34 simpr1 1065 . . . . 5 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → 𝐹:𝐵𝐶)
354lmodring 18787 . . . . . . . . . 10 (𝑆 ∈ LMod → 𝐾 ∈ Ring)
3635ad2antrr 761 . . . . . . . . 9 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) → 𝐾 ∈ Ring)
37 eqid 2626 . . . . . . . . . 10 (1r𝐾) = (1r𝐾)
3814, 37ringidcl 18484 . . . . . . . . 9 (𝐾 ∈ Ring → (1r𝐾) ∈ 𝐸)
39 oveq1 6612 . . . . . . . . . . . . . 14 (𝑥 = (1r𝐾) → (𝑥 · 𝑦) = ((1r𝐾) · 𝑦))
4039oveq1d 6620 . . . . . . . . . . . . 13 (𝑥 = (1r𝐾) → ((𝑥 · 𝑦) + 𝑧) = (((1r𝐾) · 𝑦) + 𝑧))
4140fveq2d 6154 . . . . . . . . . . . 12 (𝑥 = (1r𝐾) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)))
42 oveq1 6612 . . . . . . . . . . . . 13 (𝑥 = (1r𝐾) → (𝑥 × (𝐹𝑦)) = ((1r𝐾) × (𝐹𝑦)))
4342oveq1d 6620 . . . . . . . . . . . 12 (𝑥 = (1r𝐾) → ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧)))
4441, 43eqeq12d 2641 . . . . . . . . . . 11 (𝑥 = (1r𝐾) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) ↔ (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧))))
45442ralbidv 2988 . . . . . . . . . 10 (𝑥 = (1r𝐾) → (∀𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧))))
4645rspcv 3296 . . . . . . . . 9 ((1r𝐾) ∈ 𝐸 → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑦𝐵𝑧𝐵 (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧))))
4736, 38, 463syl 18 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑦𝐵𝑧𝐵 (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧))))
48 simplll 797 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → 𝑆 ∈ LMod)
49 simprl 793 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → 𝑦𝐵)
501, 4, 13, 37lmodvs1 18807 . . . . . . . . . . . . 13 ((𝑆 ∈ LMod ∧ 𝑦𝐵) → ((1r𝐾) · 𝑦) = 𝑦)
5148, 49, 50syl2anc 692 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → ((1r𝐾) · 𝑦) = 𝑦)
5251oveq1d 6620 . . . . . . . . . . 11 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → (((1r𝐾) · 𝑦) + 𝑧) = (𝑦 + 𝑧))
5352fveq2d 6154 . . . . . . . . . 10 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (𝐹‘(𝑦 + 𝑧)))
54 simplrr 800 . . . . . . . . . . . . . 14 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → 𝐿 = 𝐾)
5554fveq2d 6154 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → (1r𝐿) = (1r𝐾))
5655oveq1d 6620 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → ((1r𝐿) × (𝐹𝑦)) = ((1r𝐾) × (𝐹𝑦)))
57 simpllr 798 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → 𝑇 ∈ LMod)
58 simplrl 799 . . . . . . . . . . . . . 14 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → 𝐹:𝐵𝐶)
5958, 49ffvelrnd 6317 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → (𝐹𝑦) ∈ 𝐶)
60 eqid 2626 . . . . . . . . . . . . . 14 (1r𝐿) = (1r𝐿)
612, 5, 22, 60lmodvs1 18807 . . . . . . . . . . . . 13 ((𝑇 ∈ LMod ∧ (𝐹𝑦) ∈ 𝐶) → ((1r𝐿) × (𝐹𝑦)) = (𝐹𝑦))
6257, 59, 61syl2anc 692 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → ((1r𝐿) × (𝐹𝑦)) = (𝐹𝑦))
6356, 62eqtr3d 2662 . . . . . . . . . . 11 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → ((1r𝐾) × (𝐹𝑦)) = (𝐹𝑦))
6463oveq1d 6620 . . . . . . . . . 10 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧)) = ((𝐹𝑦) (𝐹𝑧)))
6553, 64eqeq12d 2641 . . . . . . . . 9 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) ∧ (𝑦𝐵𝑧𝐵)) → ((𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧)) ↔ (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧))))
66652ralbidva 2987 . . . . . . . 8 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) → (∀𝑦𝐵𝑧𝐵 (𝐹‘(((1r𝐾) · 𝑦) + 𝑧)) = (((1r𝐾) × (𝐹𝑦)) (𝐹𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧))))
6747, 66sylibd 229 . . . . . . 7 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾)) → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧))))
6867exp32 630 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵𝐶 → (𝐿 = 𝐾 → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧))))))
69683imp2 1279 . . . . 5 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧)))
7034, 69jca 554 . . . 4 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → (𝐹:𝐵𝐶 ∧ ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧))))
711, 2, 18, 19isghm 17576 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑦𝐵𝑧𝐵 (𝐹‘(𝑦 + 𝑧)) = ((𝐹𝑦) (𝐹𝑧)))))
7233, 70, 71sylanbrc 697 . . 3 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
73 simpr2 1066 . . 3 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → 𝐿 = 𝐾)
74 eqid 2626 . . . . . 6 (0g𝑆) = (0g𝑆)
75 eqid 2626 . . . . . 6 (0g𝑇) = (0g𝑇)
7674, 75ghmid 17582 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
7772, 76syl 17 . . . 4 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → (𝐹‘(0g𝑆)) = (0g𝑇))
7830ad3antrrr 765 . . . . . . . . . 10 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑆 ∈ Grp)
791, 74grpidcl 17366 . . . . . . . . . 10 (𝑆 ∈ Grp → (0g𝑆) ∈ 𝐵)
80 oveq2 6613 . . . . . . . . . . . . 13 (𝑧 = (0g𝑆) → ((𝑥 · 𝑦) + 𝑧) = ((𝑥 · 𝑦) + (0g𝑆)))
8180fveq2d 6154 . . . . . . . . . . . 12 (𝑧 = (0g𝑆) → (𝐹‘((𝑥 · 𝑦) + 𝑧)) = (𝐹‘((𝑥 · 𝑦) + (0g𝑆))))
82 fveq2 6150 . . . . . . . . . . . . 13 (𝑧 = (0g𝑆) → (𝐹𝑧) = (𝐹‘(0g𝑆)))
8382oveq2d 6621 . . . . . . . . . . . 12 (𝑧 = (0g𝑆) → ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆))))
8481, 83eqeq12d 2641 . . . . . . . . . . 11 (𝑧 = (0g𝑆) → ((𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) ↔ (𝐹‘((𝑥 · 𝑦) + (0g𝑆))) = ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆)))))
8584rspcv 3296 . . . . . . . . . 10 ((0g𝑆) ∈ 𝐵 → (∀𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → (𝐹‘((𝑥 · 𝑦) + (0g𝑆))) = ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆)))))
8678, 79, 853syl 18 . . . . . . . . 9 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (∀𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → (𝐹‘((𝑥 · 𝑦) + (0g𝑆))) = ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆)))))
87 simplll 797 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑆 ∈ LMod)
88 simprl 793 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑥𝐸)
89 simprr 795 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑦𝐵)
9087, 88, 89, 15syl3anc 1323 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (𝑥 · 𝑦) ∈ 𝐵)
911, 18, 74grprid 17369 . . . . . . . . . . . 12 ((𝑆 ∈ Grp ∧ (𝑥 · 𝑦) ∈ 𝐵) → ((𝑥 · 𝑦) + (0g𝑆)) = (𝑥 · 𝑦))
9278, 90, 91syl2anc 692 . . . . . . . . . . 11 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → ((𝑥 · 𝑦) + (0g𝑆)) = (𝑥 · 𝑦))
9392fveq2d 6154 . . . . . . . . . 10 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (𝐹‘((𝑥 · 𝑦) + (0g𝑆))) = (𝐹‘(𝑥 · 𝑦)))
94 simplr3 1103 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (𝐹‘(0g𝑆)) = (0g𝑇))
9594oveq2d 6621 . . . . . . . . . . 11 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆))) = ((𝑥 × (𝐹𝑦)) (0g𝑇)))
96 simpllr 798 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑇 ∈ LMod)
9796, 31syl 17 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑇 ∈ Grp)
98 simplr2 1102 . . . . . . . . . . . . . . . 16 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝐿 = 𝐾)
9998fveq2d 6154 . . . . . . . . . . . . . . 15 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (Base‘𝐿) = (Base‘𝐾))
10099, 14syl6eqr 2678 . . . . . . . . . . . . . 14 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (Base‘𝐿) = 𝐸)
10188, 100eleqtrrd 2707 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝑥 ∈ (Base‘𝐿))
102 simplr1 1101 . . . . . . . . . . . . . 14 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → 𝐹:𝐵𝐶)
103102, 89ffvelrnd 6317 . . . . . . . . . . . . 13 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (𝐹𝑦) ∈ 𝐶)
104 eqid 2626 . . . . . . . . . . . . . 14 (Base‘𝐿) = (Base‘𝐿)
1052, 5, 22, 104lmodvscl 18796 . . . . . . . . . . . . 13 ((𝑇 ∈ LMod ∧ 𝑥 ∈ (Base‘𝐿) ∧ (𝐹𝑦) ∈ 𝐶) → (𝑥 × (𝐹𝑦)) ∈ 𝐶)
10696, 101, 103, 105syl3anc 1323 . . . . . . . . . . . 12 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (𝑥 × (𝐹𝑦)) ∈ 𝐶)
1072, 19, 75grprid 17369 . . . . . . . . . . . 12 ((𝑇 ∈ Grp ∧ (𝑥 × (𝐹𝑦)) ∈ 𝐶) → ((𝑥 × (𝐹𝑦)) (0g𝑇)) = (𝑥 × (𝐹𝑦)))
10897, 106, 107syl2anc 692 . . . . . . . . . . 11 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → ((𝑥 × (𝐹𝑦)) (0g𝑇)) = (𝑥 × (𝐹𝑦)))
10995, 108eqtrd 2660 . . . . . . . . . 10 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆))) = (𝑥 × (𝐹𝑦)))
11093, 109eqeq12d 2641 . . . . . . . . 9 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → ((𝐹‘((𝑥 · 𝑦) + (0g𝑆))) = ((𝑥 × (𝐹𝑦)) (𝐹‘(0g𝑆))) ↔ (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
11186, 110sylibd 229 . . . . . . . 8 ((((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) ∧ (𝑥𝐸𝑦𝐵)) → (∀𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
112111ralimdvva 2963 . . . . . . 7 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ (𝐹‘(0g𝑆)) = (0g𝑇))) → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
1131123exp2 1282 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵𝐶 → (𝐿 = 𝐾 → ((𝐹‘(0g𝑆)) = (0g𝑇) → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))))
114113com45 97 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹:𝐵𝐶 → (𝐿 = 𝐾 → (∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)) → ((𝐹‘(0g𝑆)) = (0g𝑇) → ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))))
1151143imp2 1279 . . . 4 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → ((𝐹‘(0g𝑆)) = (0g𝑇) → ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦))))
11677, 115mpd 15 . . 3 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))
1174, 5, 14, 1, 13, 22islmhm3 18942 . . . 4 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
118117adantr 481 . . 3 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))
11972, 73, 116, 118mpbir3and 1243 . 2 (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
12029, 119impbida 876 1 ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Scalarcsca 15860   ·𝑠 cvsca 15861  0gc0g 16016  Grpcgrp 17338   GrpHom cghm 17573  1rcur 18417  Ringcrg 18463  LModclmod 18779   LMHom clmhm 18933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-ghm 17574  df-mgp 18406  df-ur 18418  df-ring 18465  df-lmod 18781  df-lmhm 18936
This theorem is referenced by:  isphld  19913
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