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Theorem mrsubco 34543
Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mrsubco.s 𝑆 = (mRSubstβ€˜π‘‡)
Assertion
Ref Expression
mrsubco ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Proof of Theorem mrsubco
Dummy variables 𝑐 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrsubco.s . . . . 5 𝑆 = (mRSubstβ€˜π‘‡)
2 eqid 2733 . . . . 5 (mRExβ€˜π‘‡) = (mRExβ€˜π‘‡)
31, 2mrsubf 34539 . . . 4 (𝐹 ∈ ran 𝑆 β†’ 𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
43adantr 482 . . 3 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
51, 2mrsubf 34539 . . . 4 (𝐺 ∈ ran 𝑆 β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
65adantl 483 . . 3 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
7 fco 6742 . . 3 ((𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡)) β†’ (𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
84, 6, 7syl2anc 585 . 2 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
96adantr 482 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
10 eldifi 4127 . . . . . . . . 9 (𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡)) β†’ 𝑐 ∈ (mCNβ€˜π‘‡))
11 elun1 4177 . . . . . . . . 9 (𝑐 ∈ (mCNβ€˜π‘‡) β†’ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
1210, 11syl 17 . . . . . . . 8 (𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡)) β†’ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
1312adantl 483 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
1413s1cld 14553 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ βŸ¨β€œπ‘β€βŸ© ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
15 n0i 4334 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 β†’ Β¬ ran 𝑆 = βˆ…)
161rnfvprc 6886 . . . . . . . . . 10 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 = βˆ…)
1715, 16nsyl2 141 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 β†’ 𝑇 ∈ V)
1817adantr 482 . . . . . . . 8 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝑇 ∈ V)
1918adantr 482 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝑇 ∈ V)
20 eqid 2733 . . . . . . . 8 (mCNβ€˜π‘‡) = (mCNβ€˜π‘‡)
21 eqid 2733 . . . . . . . 8 (mVRβ€˜π‘‡) = (mVRβ€˜π‘‡)
2220, 21, 2mrexval 34523 . . . . . . 7 (𝑇 ∈ V β†’ (mRExβ€˜π‘‡) = Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
2319, 22syl 17 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (mRExβ€˜π‘‡) = Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
2414, 23eleqtrrd 2837 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ βŸ¨β€œπ‘β€βŸ© ∈ (mRExβ€˜π‘‡))
25 fvco3 6991 . . . . 5 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ βŸ¨β€œπ‘β€βŸ© ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = (πΉβ€˜(πΊβ€˜βŸ¨β€œπ‘β€βŸ©)))
269, 24, 25syl2anc 585 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = (πΉβ€˜(πΊβ€˜βŸ¨β€œπ‘β€βŸ©)))
271, 2, 21, 20mrsubcn 34541 . . . . . 6 ((𝐺 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΊβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
2827adantll 713 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΊβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
2928fveq2d 6896 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΉβ€˜(πΊβ€˜βŸ¨β€œπ‘β€βŸ©)) = (πΉβ€˜βŸ¨β€œπ‘β€βŸ©))
301, 2, 21, 20mrsubcn 34541 . . . . 5 ((𝐹 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΉβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
3130adantlr 714 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΉβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
3226, 29, 313eqtrd 2777 . . 3 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
3332ralrimiva 3147 . 2 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ βˆ€π‘ ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
341, 2mrsubccat 34540 . . . . . . . 8 ((𝐺 ∈ ran 𝑆 ∧ π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡)) β†’ (πΊβ€˜(π‘₯ ++ 𝑦)) = ((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦)))
35343expb 1121 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜(π‘₯ ++ 𝑦)) = ((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦)))
3635adantll 713 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜(π‘₯ ++ 𝑦)) = ((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦)))
3736fveq2d 6896 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΉβ€˜(πΊβ€˜(π‘₯ ++ 𝑦))) = (πΉβ€˜((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦))))
38 simpll 766 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝐹 ∈ ran 𝑆)
396adantr 482 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
40 simprl 770 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ π‘₯ ∈ (mRExβ€˜π‘‡))
4139, 40ffvelcdmd 7088 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜π‘₯) ∈ (mRExβ€˜π‘‡))
42 simprr 772 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝑦 ∈ (mRExβ€˜π‘‡))
4339, 42ffvelcdmd 7088 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜π‘¦) ∈ (mRExβ€˜π‘‡))
441, 2mrsubccat 34540 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (πΊβ€˜π‘₯) ∈ (mRExβ€˜π‘‡) ∧ (πΊβ€˜π‘¦) ∈ (mRExβ€˜π‘‡)) β†’ (πΉβ€˜((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦))) = ((πΉβ€˜(πΊβ€˜π‘₯)) ++ (πΉβ€˜(πΊβ€˜π‘¦))))
4538, 41, 43, 44syl3anc 1372 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΉβ€˜((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦))) = ((πΉβ€˜(πΊβ€˜π‘₯)) ++ (πΉβ€˜(πΊβ€˜π‘¦))))
4637, 45eqtrd 2773 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΉβ€˜(πΊβ€˜(π‘₯ ++ 𝑦))) = ((πΉβ€˜(πΊβ€˜π‘₯)) ++ (πΉβ€˜(πΊβ€˜π‘¦))))
4718, 22syl 17 . . . . . . . . 9 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (mRExβ€˜π‘‡) = Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
4847adantr 482 . . . . . . . 8 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (mRExβ€˜π‘‡) = Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
4940, 48eleqtrd 2836 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ π‘₯ ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
5042, 48eleqtrd 2836 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝑦 ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
51 ccatcl 14524 . . . . . . 7 ((π‘₯ ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)) ∧ 𝑦 ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡))) β†’ (π‘₯ ++ 𝑦) ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
5249, 50, 51syl2anc 585 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (π‘₯ ++ 𝑦) ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
5352, 48eleqtrrd 2837 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (π‘₯ ++ 𝑦) ∈ (mRExβ€˜π‘‡))
54 fvco3 6991 . . . . 5 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ (π‘₯ ++ 𝑦) ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (πΉβ€˜(πΊβ€˜(π‘₯ ++ 𝑦))))
5539, 53, 54syl2anc 585 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (πΉβ€˜(πΊβ€˜(π‘₯ ++ 𝑦))))
56 fvco3 6991 . . . . . 6 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ π‘₯ ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
5739, 40, 56syl2anc 585 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
58 fvco3 6991 . . . . . 6 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
5939, 42, 58syl2anc 585 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
6057, 59oveq12d 7427 . . . 4 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (((𝐹 ∘ 𝐺)β€˜π‘₯) ++ ((𝐹 ∘ 𝐺)β€˜π‘¦)) = ((πΉβ€˜(πΊβ€˜π‘₯)) ++ (πΉβ€˜(πΊβ€˜π‘¦))))
6146, 55, 603eqtr4d 2783 . . 3 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ ((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (((𝐹 ∘ 𝐺)β€˜π‘₯) ++ ((𝐹 ∘ 𝐺)β€˜π‘¦)))
6261ralrimivva 3201 . 2 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ βˆ€π‘₯ ∈ (mRExβ€˜π‘‡)βˆ€π‘¦ ∈ (mRExβ€˜π‘‡)((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (((𝐹 ∘ 𝐺)β€˜π‘₯) ++ ((𝐹 ∘ 𝐺)β€˜π‘¦)))
631, 2, 21, 20elmrsubrn 34542 . . 3 (𝑇 ∈ V β†’ ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ βˆ€π‘ ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ© ∧ βˆ€π‘₯ ∈ (mRExβ€˜π‘‡)βˆ€π‘¦ ∈ (mRExβ€˜π‘‡)((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (((𝐹 ∘ 𝐺)β€˜π‘₯) ++ ((𝐹 ∘ 𝐺)β€˜π‘¦)))))
6418, 63syl 17 . 2 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ βˆ€π‘ ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))((𝐹 ∘ 𝐺)β€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ© ∧ βˆ€π‘₯ ∈ (mRExβ€˜π‘‡)βˆ€π‘¦ ∈ (mRExβ€˜π‘‡)((𝐹 ∘ 𝐺)β€˜(π‘₯ ++ 𝑦)) = (((𝐹 ∘ 𝐺)β€˜π‘₯) ++ ((𝐹 ∘ 𝐺)β€˜π‘¦)))))
658, 33, 62, 64mpbir3and 1343 1 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   βˆ– cdif 3946   βˆͺ cun 3947  βˆ…c0 4323  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Word cword 14464   ++ cconcat 14520  βŸ¨β€œcs1 14545  mCNcmcn 34482  mVRcmvar 34483  mRExcmrex 34488  mRSubstcmrsub 34492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-seq 13967  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-0g 17387  df-gsum 17388  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-frmd 18730  df-vrmd 18731  df-mrex 34508  df-mrsub 34512
This theorem is referenced by:  msubco  34553
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