Theorem ixpsnbasval 14504

Description: The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)

Assertion

Ref	Expression
ixpsnbasval	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))})

Distinct variable groups:   𝑅,𝑓,𝑥   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋,𝑥

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ixpsnbasval

Step	Hyp	Ref	Expression
1		ixpsnval 6875	. . 3 ⊢ (𝑋 ∈ 𝑊 → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)))})
2	1	adantl 277	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)))})
3		rlmfn 14491	. . . . . . . . . . . 12 ⊢ ringLMod Fn V
4		elex 2813	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
5		funfvex 5659	. . . . . . . . . . . . 13 ⊢ ((Fun ringLMod ∧ 𝑅 ∈ dom ringLMod) → (ringLMod‘𝑅) ∈ V)
6	5	funfni 5434	. . . . . . . . . . . 12 ⊢ ((ringLMod Fn V ∧ 𝑅 ∈ V) → (ringLMod‘𝑅) ∈ V)
7	3, 4, 6	sylancr 414	. . . . . . . . . . 11 ⊢ (𝑅 ∈ 𝑉 → (ringLMod‘𝑅) ∈ V)
8	7	anim1ci 341	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ 𝑊 ∧ (ringLMod‘𝑅) ∈ V))
9		xpsng 5826	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑊 ∧ (ringLMod‘𝑅) ∈ V) → ({𝑋} × {(ringLMod‘𝑅)}) = {⟨𝑋, (ringLMod‘𝑅)⟩})
10	8, 9	syl 14	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ({𝑋} × {(ringLMod‘𝑅)}) = {⟨𝑋, (ringLMod‘𝑅)⟩})
11	10	fveq1d 5644	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (({𝑋} × {(ringLMod‘𝑅)})‘𝑋) = ({⟨𝑋, (ringLMod‘𝑅)⟩}‘𝑋))
12		fvsng 5853	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (ringLMod‘𝑅) ∈ V) → ({⟨𝑋, (ringLMod‘𝑅)⟩}‘𝑋) = (ringLMod‘𝑅))
13	8, 12	syl 14	. . . . . . . 8 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ({⟨𝑋, (ringLMod‘𝑅)⟩}‘𝑋) = (ringLMod‘𝑅))
14	11, 13	eqtrd 2263	. . . . . . 7 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (({𝑋} × {(ringLMod‘𝑅)})‘𝑋) = (ringLMod‘𝑅))
15	14	fveq2d 5646	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑋)) = (Base‘(ringLMod‘𝑅)))
16		csbfv2g 5683	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = (Base‘⦋𝑋 / 𝑥⦌(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)))
17		csbfvg 5684	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑊 → ⦋𝑋 / 𝑥⦌(({𝑋} × {(ringLMod‘𝑅)})‘𝑥) = (({𝑋} × {(ringLMod‘𝑅)})‘𝑋))
18	17	fveq2d 5646	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑊 → (Base‘⦋𝑋 / 𝑥⦌(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑋)))
19	16, 18	eqtrd 2263	. . . . . . 7 ⊢ (𝑋 ∈ 𝑊 → ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑋)))
20	19	adantl 277	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑋)))
21		rlmbasg 14493	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘(ringLMod‘𝑅)))
22	21	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (Base‘𝑅) = (Base‘(ringLMod‘𝑅)))
23	15, 20, 22	3eqtr4d 2273	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = (Base‘𝑅))
24	23	eleq2d 2300	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ((𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) ↔ (𝑓‘𝑋) ∈ (Base‘𝑅)))
25	24	anbi2d 464	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ((𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥))) ↔ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))))
26	25	abbidv 2348	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌(Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)))} = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))})
27	2, 26	eqtrd 2263	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → X𝑥 ∈ {𝑋} (Base‘(({𝑋} × {(ringLMod‘𝑅)})‘𝑥)) = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ (Base‘𝑅))})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  {cab 2216  Vcvv 2801  ⦋csb 3126  {csn 3670  ⟨cop 3673   × cxp 4725   Fn wfn 5323  ‘cfv 5328  Xcixp 6872  Basecbs 13105  ringLModcrglmod 14472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-ixp 6873  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-sra 14473  df-rgmod 14474

This theorem is referenced by: (None)