Theorem divsmulw 28047

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28071, we can eliminate the existence hypothesis (see divsmul 28074). (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
divsmulw	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem divsmulw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divsval 28044	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → (𝐴 /su 𝐶) = (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴))
2	1	eqeq1d 2728	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
3	2	3expb 1117	. . . 4 ⊢ ((𝐴 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
4	3	3adant2 1128	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
5	4	adantr 480	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
6		simpl2 1189	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → 𝐵 ∈ No )
7		simp3l 1198	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ∈ No )
8		simp3r 1199	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ≠ 0s )
9		simp1 1133	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐴 ∈ No )
10	7, 8, 9	3jca 1125	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → (𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ))
11		noreceuw 28046	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No ) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
12	10, 11	sylan 579	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
13		oveq2 7413	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 ·s 𝑦) = (𝐶 ·s 𝐵))
14	13	eqeq1d 2728	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐶 ·s 𝑦) = 𝐴 ↔ (𝐶 ·s 𝐵) = 𝐴))
15	14	riota2 7387	. . 3 ⊢ ((𝐵 ∈ No ∧ ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
16	6, 12, 15	syl2anc 583	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
17	5, 16	bitr4d 282	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∃wrex 3064  ∃!wreu 3368  ℩crio 7360  (class class class)co 7405   No csur 27528   0_s c0s 27710   1_s c1s 27711   ·_s cmuls 27961   /_su cdivs 28042

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

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-ral 3056  df-rex 3065  df-rmo 3370  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-ot 4632  df-uni 4903  df-int 4944  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-se 5625  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-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-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-1o 8467  df-2o 8468  df-nadd 8667  df-no 27531  df-slt 27532  df-bday 27533  df-sle 27633  df-sslt 27669  df-scut 27671  df-0s 27712  df-1s 27713  df-made 27729  df-old 27730  df-left 27732  df-right 27733  df-norec 27810  df-norec2 27821  df-adds 27832  df-negs 27889  df-subs 27890  df-muls 27962  df-divs 28043

This theorem is referenced by:  divsmulwd  28048  divs1  28058  divsmul  28074