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Theorem rngccoALTV 47226
Description: Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
rngcbasALTV.c 𝐢 = (RngCatALTVβ€˜π‘ˆ)
rngcbasALTV.b 𝐡 = (Baseβ€˜πΆ)
rngcbasALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngccofvalALTV.o Β· = (compβ€˜πΆ)
rngccoALTV.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
rngccoALTV.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
rngccoALTV.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
rngccoALTV.f (πœ‘ β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
rngccoALTV.g (πœ‘ β†’ 𝐺 ∈ (π‘Œ RngHom 𝑍))
Assertion
Ref Expression
rngccoALTV (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))

Proof of Theorem rngccoALTV
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcbasALTV.c . . . 4 𝐢 = (RngCatALTVβ€˜π‘ˆ)
2 rngcbasALTV.b . . . 4 𝐡 = (Baseβ€˜πΆ)
3 rngcbasALTV.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 rngccofvalALTV.o . . . 4 Β· = (compβ€˜πΆ)
51, 2, 3, 4rngccofvalALTV 47225 . . 3 (πœ‘ β†’ Β· = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
6 simprl 768 . . . . . . 7 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑣 = βŸ¨π‘‹, π‘ŒβŸ©)
76fveq2d 6889 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©))
8 rngccoALTV.x . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ 𝐡)
9 rngccoALTV.y . . . . . . . 8 (πœ‘ β†’ π‘Œ ∈ 𝐡)
10 op2ndg 7987 . . . . . . . 8 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
118, 9, 10syl2anc 583 . . . . . . 7 (πœ‘ β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
1211adantr 480 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜βŸ¨π‘‹, π‘ŒβŸ©) = π‘Œ)
137, 12eqtrd 2766 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (2nd β€˜π‘£) = π‘Œ)
14 simprr 770 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ 𝑧 = 𝑍)
1513, 14oveq12d 7423 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((2nd β€˜π‘£) RngHom 𝑧) = (π‘Œ RngHom 𝑍))
166fveq2d 6889 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©))
17 op1stg 7986 . . . . . . . 8 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
188, 9, 17syl2anc 583 . . . . . . 7 (πœ‘ β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
1918adantr 480 . . . . . 6 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜βŸ¨π‘‹, π‘ŒβŸ©) = 𝑋)
2016, 19eqtrd 2766 . . . . 5 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (1st β€˜π‘£) = 𝑋)
2120, 13oveq12d 7423 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) = (𝑋 RngHom π‘Œ))
22 eqidd 2727 . . . 4 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∘ 𝑓) = (𝑔 ∘ 𝑓))
2315, 21, 22mpoeq123dv 7480 . . 3 ((πœ‘ ∧ (𝑣 = βŸ¨π‘‹, π‘ŒβŸ© ∧ 𝑧 = 𝑍)) β†’ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (π‘Œ RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom π‘Œ) ↦ (𝑔 ∘ 𝑓)))
24 opelxpi 5706 . . . 4 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
258, 9, 24syl2anc 583 . . 3 (πœ‘ β†’ βŸ¨π‘‹, π‘ŒβŸ© ∈ (𝐡 Γ— 𝐡))
26 rngccoALTV.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
27 ovex 7438 . . . . 5 (π‘Œ RngHom 𝑍) ∈ V
28 ovex 7438 . . . . 5 (𝑋 RngHom π‘Œ) ∈ V
2927, 28mpoex 8065 . . . 4 (𝑔 ∈ (π‘Œ RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom π‘Œ) ↦ (𝑔 ∘ 𝑓)) ∈ V
3029a1i 11 . . 3 (πœ‘ β†’ (𝑔 ∈ (π‘Œ RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom π‘Œ) ↦ (𝑔 ∘ 𝑓)) ∈ V)
315, 23, 25, 26, 30ovmpod 7556 . 2 (πœ‘ β†’ (βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍) = (𝑔 ∈ (π‘Œ RngHom 𝑍), 𝑓 ∈ (𝑋 RngHom π‘Œ) ↦ (𝑔 ∘ 𝑓)))
32 simprl 768 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑔 = 𝐺)
33 simprr 770 . . 3 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ 𝑓 = 𝐹)
3432, 33coeq12d 5858 . 2 ((πœ‘ ∧ (𝑔 = 𝐺 ∧ 𝑓 = 𝐹)) β†’ (𝑔 ∘ 𝑓) = (𝐺 ∘ 𝐹))
35 rngccoALTV.g . 2 (πœ‘ β†’ 𝐺 ∈ (π‘Œ RngHom 𝑍))
36 rngccoALTV.f . 2 (πœ‘ β†’ 𝐹 ∈ (𝑋 RngHom π‘Œ))
37 coexg 7919 . . 3 ((𝐺 ∈ (π‘Œ RngHom 𝑍) ∧ 𝐹 ∈ (𝑋 RngHom π‘Œ)) β†’ (𝐺 ∘ 𝐹) ∈ V)
3835, 36, 37syl2anc 583 . 2 (πœ‘ β†’ (𝐺 ∘ 𝐹) ∈ V)
3931, 34, 35, 36, 38ovmpod 7556 1 (πœ‘ β†’ (𝐺(βŸ¨π‘‹, π‘ŒβŸ© Β· 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βŸ¨cop 4629   Γ— cxp 5667   ∘ ccom 5673  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7972  2nd c2nd 7973  Basecbs 17153  compcco 17218   RngHom crnghm 20336  RngCatALTVcrngcALTV 47218
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-hom 17230  df-cco 17231  df-rngcALTV 47219
This theorem is referenced by:  rngccatidALTV  47227  rngcsectALTV  47230  rhmsubcALTVlem4  47239
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