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Theorem prdsmgp 19360
Description: The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
Hypotheses
Ref Expression
prdsmgp.y 𝑌 = (𝑆Xs𝑅)
prdsmgp.m 𝑀 = (mulGrp‘𝑌)
prdsmgp.z 𝑍 = (𝑆Xs(mulGrp ∘ 𝑅))
prdsmgp.i (𝜑𝐼𝑉)
prdsmgp.s (𝜑𝑆𝑊)
prdsmgp.r (𝜑𝑅 Fn 𝐼)
Assertion
Ref Expression
prdsmgp (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g𝑀) = (+g𝑍)))

Proof of Theorem prdsmgp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . . . 6 (mulGrp‘(𝑅𝑥)) = (mulGrp‘(𝑅𝑥))
2 eqid 2801 . . . . . 6 (Base‘(𝑅𝑥)) = (Base‘(𝑅𝑥))
31, 2mgpbas 19242 . . . . 5 (Base‘(𝑅𝑥)) = (Base‘(mulGrp‘(𝑅𝑥)))
4 prdsmgp.r . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
5 fvco2 6739 . . . . . . . 8 ((𝑅 Fn 𝐼𝑥𝐼) → ((mulGrp ∘ 𝑅)‘𝑥) = (mulGrp‘(𝑅𝑥)))
64, 5sylan 583 . . . . . . 7 ((𝜑𝑥𝐼) → ((mulGrp ∘ 𝑅)‘𝑥) = (mulGrp‘(𝑅𝑥)))
76eqcomd 2807 . . . . . 6 ((𝜑𝑥𝐼) → (mulGrp‘(𝑅𝑥)) = ((mulGrp ∘ 𝑅)‘𝑥))
87fveq2d 6653 . . . . 5 ((𝜑𝑥𝐼) → (Base‘(mulGrp‘(𝑅𝑥))) = (Base‘((mulGrp ∘ 𝑅)‘𝑥)))
93, 8syl5eq 2848 . . . 4 ((𝜑𝑥𝐼) → (Base‘(𝑅𝑥)) = (Base‘((mulGrp ∘ 𝑅)‘𝑥)))
109ixpeq2dva 8463 . . 3 (𝜑X𝑥𝐼 (Base‘(𝑅𝑥)) = X𝑥𝐼 (Base‘((mulGrp ∘ 𝑅)‘𝑥)))
11 prdsmgp.y . . . 4 𝑌 = (𝑆Xs𝑅)
12 prdsmgp.m . . . . . 6 𝑀 = (mulGrp‘𝑌)
13 eqid 2801 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
1412, 13mgpbas 19242 . . . . 5 (Base‘𝑌) = (Base‘𝑀)
1514eqcomi 2810 . . . 4 (Base‘𝑀) = (Base‘𝑌)
16 prdsmgp.s . . . 4 (𝜑𝑆𝑊)
17 prdsmgp.i . . . 4 (𝜑𝐼𝑉)
1811, 15, 16, 17, 4prdsbas2 16738 . . 3 (𝜑 → (Base‘𝑀) = X𝑥𝐼 (Base‘(𝑅𝑥)))
19 prdsmgp.z . . . 4 𝑍 = (𝑆Xs(mulGrp ∘ 𝑅))
20 eqid 2801 . . . 4 (Base‘𝑍) = (Base‘𝑍)
21 fnmgp 19238 . . . . 5 mulGrp Fn V
22 ssv 3942 . . . . . 6 ran 𝑅 ⊆ V
2322a1i 11 . . . . 5 (𝜑 → ran 𝑅 ⊆ V)
24 fnco 6441 . . . . 5 ((mulGrp Fn V ∧ 𝑅 Fn 𝐼 ∧ ran 𝑅 ⊆ V) → (mulGrp ∘ 𝑅) Fn 𝐼)
2521, 4, 23, 24mp3an2i 1463 . . . 4 (𝜑 → (mulGrp ∘ 𝑅) Fn 𝐼)
2619, 20, 16, 17, 25prdsbas2 16738 . . 3 (𝜑 → (Base‘𝑍) = X𝑥𝐼 (Base‘((mulGrp ∘ 𝑅)‘𝑥)))
2710, 18, 263eqtr4d 2846 . 2 (𝜑 → (Base‘𝑀) = (Base‘𝑍))
28 eqid 2801 . . . 4 (.r𝑌) = (.r𝑌)
2912, 28mgpplusg 19240 . . 3 (.r𝑌) = (+g𝑀)
30 eqid 2801 . . . . . . . . 9 (mulGrp‘(𝑅𝑧)) = (mulGrp‘(𝑅𝑧))
31 eqid 2801 . . . . . . . . 9 (.r‘(𝑅𝑧)) = (.r‘(𝑅𝑧))
3230, 31mgpplusg 19240 . . . . . . . 8 (.r‘(𝑅𝑧)) = (+g‘(mulGrp‘(𝑅𝑧)))
33 fvco2 6739 . . . . . . . . . . 11 ((𝑅 Fn 𝐼𝑧𝐼) → ((mulGrp ∘ 𝑅)‘𝑧) = (mulGrp‘(𝑅𝑧)))
344, 33sylan 583 . . . . . . . . . 10 ((𝜑𝑧𝐼) → ((mulGrp ∘ 𝑅)‘𝑧) = (mulGrp‘(𝑅𝑧)))
3534eqcomd 2807 . . . . . . . . 9 ((𝜑𝑧𝐼) → (mulGrp‘(𝑅𝑧)) = ((mulGrp ∘ 𝑅)‘𝑧))
3635fveq2d 6653 . . . . . . . 8 ((𝜑𝑧𝐼) → (+g‘(mulGrp‘(𝑅𝑧))) = (+g‘((mulGrp ∘ 𝑅)‘𝑧)))
3732, 36syl5eq 2848 . . . . . . 7 ((𝜑𝑧𝐼) → (.r‘(𝑅𝑧)) = (+g‘((mulGrp ∘ 𝑅)‘𝑧)))
3837oveqd 7156 . . . . . 6 ((𝜑𝑧𝐼) → ((𝑥𝑧)(.r‘(𝑅𝑧))(𝑦𝑧)) = ((𝑥𝑧)(+g‘((mulGrp ∘ 𝑅)‘𝑧))(𝑦𝑧)))
3938mpteq2dva 5128 . . . . 5 (𝜑 → (𝑧𝐼 ↦ ((𝑥𝑧)(.r‘(𝑅𝑧))(𝑦𝑧))) = (𝑧𝐼 ↦ ((𝑥𝑧)(+g‘((mulGrp ∘ 𝑅)‘𝑧))(𝑦𝑧))))
4027, 27, 39mpoeq123dv 7212 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑧𝐼 ↦ ((𝑥𝑧)(.r‘(𝑅𝑧))(𝑦𝑧)))) = (𝑥 ∈ (Base‘𝑍), 𝑦 ∈ (Base‘𝑍) ↦ (𝑧𝐼 ↦ ((𝑥𝑧)(+g‘((mulGrp ∘ 𝑅)‘𝑧))(𝑦𝑧)))))
41 fnex 6961 . . . . . 6 ((𝑅 Fn 𝐼𝐼𝑉) → 𝑅 ∈ V)
424, 17, 41syl2anc 587 . . . . 5 (𝜑𝑅 ∈ V)
434fndmd 6431 . . . . 5 (𝜑 → dom 𝑅 = 𝐼)
4411, 16, 42, 15, 43, 28prdsmulr 16728 . . . 4 (𝜑 → (.r𝑌) = (𝑥 ∈ (Base‘𝑀), 𝑦 ∈ (Base‘𝑀) ↦ (𝑧𝐼 ↦ ((𝑥𝑧)(.r‘(𝑅𝑧))(𝑦𝑧)))))
45 fnex 6961 . . . . . 6 (((mulGrp ∘ 𝑅) Fn 𝐼𝐼𝑉) → (mulGrp ∘ 𝑅) ∈ V)
4625, 17, 45syl2anc 587 . . . . 5 (𝜑 → (mulGrp ∘ 𝑅) ∈ V)
4725fndmd 6431 . . . . 5 (𝜑 → dom (mulGrp ∘ 𝑅) = 𝐼)
48 eqid 2801 . . . . 5 (+g𝑍) = (+g𝑍)
4919, 16, 46, 20, 47, 48prdsplusg 16727 . . . 4 (𝜑 → (+g𝑍) = (𝑥 ∈ (Base‘𝑍), 𝑦 ∈ (Base‘𝑍) ↦ (𝑧𝐼 ↦ ((𝑥𝑧)(+g‘((mulGrp ∘ 𝑅)‘𝑧))(𝑦𝑧)))))
5040, 44, 493eqtr4d 2846 . . 3 (𝜑 → (.r𝑌) = (+g𝑍))
5129, 50syl5eqr 2850 . 2 (𝜑 → (+g𝑀) = (+g𝑍))
5227, 51jca 515 1 (𝜑 → ((Base‘𝑀) = (Base‘𝑍) ∧ (+g𝑀) = (+g𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  Vcvv 3444  wss 3884  cmpt 5113  ran crn 5524  ccom 5527   Fn wfn 6323  cfv 6328  (class class class)co 7139  cmpo 7141  Xcixp 8448  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  Xscprds 16715  mulGrpcmgp 19236
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-plusg 16574  df-mulr 16575  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-hom 16585  df-cco 16586  df-prds 16717  df-mgp 19237
This theorem is referenced by:  prdsringd  19362  prdscrngd  19363  prds1  19364  pwsmgp  19368
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