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Theorem mrsubco 34507
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 2732 . . . . 5 (mRExβ€˜π‘‡) = (mRExβ€˜π‘‡)
31, 2mrsubf 34503 . . . 4 (𝐹 ∈ ran 𝑆 β†’ 𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
43adantr 481 . . 3 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
51, 2mrsubf 34503 . . . 4 (𝐺 ∈ ran 𝑆 β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
65adantl 482 . . 3 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
7 fco 6741 . . 3 ((𝐹:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡)) β†’ (𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
84, 6, 7syl2anc 584 . 2 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺):(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
96adantr 481 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
10 eldifi 4126 . . . . . . . . 9 (𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡)) β†’ 𝑐 ∈ (mCNβ€˜π‘‡))
11 elun1 4176 . . . . . . . . 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 482 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
1413s1cld 14552 . . . . . 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 4333 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 β†’ Β¬ ran 𝑆 = βˆ…)
161rnfvprc 6885 . . . . . . . . . 10 (Β¬ 𝑇 ∈ V β†’ ran 𝑆 = βˆ…)
1715, 16nsyl2 141 . . . . . . . . 9 (𝐹 ∈ ran 𝑆 β†’ 𝑇 ∈ V)
1817adantr 481 . . . . . . . 8 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ 𝑇 ∈ V)
1918adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ 𝑇 ∈ V)
20 eqid 2732 . . . . . . . 8 (mCNβ€˜π‘‡) = (mCNβ€˜π‘‡)
21 eqid 2732 . . . . . . . 8 (mVRβ€˜π‘‡) = (mVRβ€˜π‘‡)
2220, 21, 2mrexval 34487 . . . . . . 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 2836 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ βŸ¨β€œπ‘β€βŸ© ∈ (mRExβ€˜π‘‡))
25 fvco3 6990 . . . . 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 584 . . . 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 34505 . . . . . 6 ((𝐺 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΊβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
2827adantll 712 . . . . 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 6895 . . . 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 34505 . . . . 5 ((𝐹 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCNβ€˜π‘‡) βˆ– (mVRβ€˜π‘‡))) β†’ (πΉβ€˜βŸ¨β€œπ‘β€βŸ©) = βŸ¨β€œπ‘β€βŸ©)
3130adantlr 713 . . . 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 2776 . . 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 3146 . 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 34504 . . . . . . . 8 ((𝐺 ∈ ran 𝑆 ∧ π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡)) β†’ (πΊβ€˜(π‘₯ ++ 𝑦)) = ((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦)))
35343expb 1120 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜(π‘₯ ++ 𝑦)) = ((πΊβ€˜π‘₯) ++ (πΊβ€˜π‘¦)))
3635adantll 712 . . . . . 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 6895 . . . . 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 765 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝐹 ∈ ran 𝑆)
396adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
40 simprl 769 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ π‘₯ ∈ (mRExβ€˜π‘‡))
4139, 40ffvelcdmd 7087 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜π‘₯) ∈ (mRExβ€˜π‘‡))
42 simprr 771 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝑦 ∈ (mRExβ€˜π‘‡))
4339, 42ffvelcdmd 7087 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (πΊβ€˜π‘¦) ∈ (mRExβ€˜π‘‡))
441, 2mrsubccat 34504 . . . . . 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 1371 . . . . 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 2772 . . . 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 481 . . . . . . . 8 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (mRExβ€˜π‘‡) = Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
4940, 48eleqtrd 2835 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ π‘₯ ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
5042, 48eleqtrd 2835 . . . . . . 7 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ 𝑦 ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
51 ccatcl 14523 . . . . . . 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 584 . . . . . 6 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (π‘₯ ++ 𝑦) ∈ Word ((mCNβ€˜π‘‡) βˆͺ (mVRβ€˜π‘‡)))
5352, 48eleqtrrd 2836 . . . . 5 (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (π‘₯ ∈ (mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡))) β†’ (π‘₯ ++ 𝑦) ∈ (mRExβ€˜π‘‡))
54 fvco3 6990 . . . . 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 584 . . . 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 6990 . . . . . 6 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ π‘₯ ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
5739, 40, 56syl2anc 584 . . . . 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 6990 . . . . . 6 ((𝐺:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ 𝑦 ∈ (mRExβ€˜π‘‡)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘¦) = (πΉβ€˜(πΊβ€˜π‘¦)))
5939, 42, 58syl2anc 584 . . . . 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 7426 . . . 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 2782 . . 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 3200 . 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 34506 . . 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 1342 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3945   βˆͺ cun 3946  βˆ…c0 4322  ran crn 5677   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Word cword 14463   ++ cconcat 14519  βŸ¨β€œcs1 14544  mCNcmcn 34446  mVRcmvar 34447  mRExcmrex 34452  mRSubstcmrsub 34456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-seq 13966  df-hash 14290  df-word 14464  df-lsw 14512  df-concat 14520  df-s1 14545  df-substr 14590  df-pfx 14620  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-gsum 17387  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-frmd 18729  df-vrmd 18730  df-mrex 34472  df-mrsub 34476
This theorem is referenced by:  msubco  34517
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